{"id":279,"date":"2016-12-08T02:44:45","date_gmt":"2016-12-08T02:44:45","guid":{"rendered":"https:\/\/blog.vanabel.cn\/?p=279"},"modified":"2017-12-26T01:39:21","modified_gmt":"2017-12-26T01:39:21","slug":"fernando-coda-marquesyuandre-neves-jiejuewillmorecaixiang","status":"publish","type":"post","link":"https:\/\/blog.vanabel.cn\/?p=279","title":{"rendered":"Fernando Coda Marques\u4e0eAndre Neves \u89e3\u51b3Willmore\u731c\u60f3"},"content":{"rendered":"<p>\u7533\u660e: \u672c\u6587\u8f6c\u81ea<a href=\"http:\/\/matheuscmss.wordpress.com\" target=\"_blank\">Matheus\u2019 Weblog<\/a>, \u539f\u6587\u9898\u76ee\u4e3a<em><a href=\"http:\/\/matheuscmss.wordpress.com\/2012\/04\/23\/the-willmore-conjecture-after-fernando-coda-marques-and-andre-neves\/\" target=\"_blank\">The Willmore conjecture after Fernando Coda Marques and Andre Neves<\/a><\/em>, \u539f\u6587\u4f5c\u8005:matheuscmss. \u5148\u7ffb\u8bd1\u81f3\u6b64, \u5e0c\u671b\u6709\u66f4\u591a\u7684\u4eba\u611f\u5174\u8da3.<\/p>\n<p>\u5728\u8fc7\u53bb\u76845\u4e2a\u6708\u4e2d, \u6211\u5f88\u9ad8\u5174\u5730\u770b\u5230\u6709\u4eba\u5ba3\u79f0\u89e3\u51b3\u4e86\u6570\u5b66\u51e0\u4e2a\u9886\u57df\u4e2d\u7684\u91cd\u8981\u7684\u95ee\u9898\u548c\u731c\u60f3.\u4f8b\u5982:<br \/>\n<ol> \t<li>Ian Agol \u5ba3\u79f0\u8bc1\u660e\u4e863\u7ef4\u62d3\u6251\u4e2d\u7684 virtual Haken \u731c\u60f3 (\u53c2\u8003 <a href=\"http:\/\/lamington.wordpress.com\/\" target=\"_blank\">D. Calegari \u535a\u5ba2<\/a>\u7684\u4e09\u7bc7\u65e5\u5fd7).<br \/>\n \t<\/li><li><a href=\"http:\/\/arxiv.org\/abs\/1202.6036\" target=\"_blank\">Fernando Cod\u00e1 Marques and Andr\u00e9 Neves<\/a> \u5ba3\u79f0\u8bc1\u660e\u4e86 <a href=\"http:\/\/en.wikipedia.org\/wiki\/Willmore_conjecture\" target=\"_blank\">Willmore \u731c\u60f3<\/a> (\u53c2\u8003 F.Morgan \u7684<a href=\"http:\/\/www.huffingtonpost.com\/frank-morgan\/math-finds-the-best-dough_b_1331844.html\" target=\"_blank\">\u8fd9\u7bc7\u6587\u7ae0<\/a>&#8212;\u975e\u6b63\u5f0f\u7684\u9610\u8ff0\u4e86\u8fd9\u4e0047\u5e74\u4e4b\u4e45\u7684\u731c\u60f3)<br \/>\n \t<\/li><li><a href=\"http:\/\/arxiv.org\/abs\/1112.5872\" target=\"_blank\">Alex Eskin, Maxim Kontsevich and Anton Zorich<\/a> \u6700\u8fd1\u5b8c\u6210\u4e86\u5173\u4e8eKontsevich-Zorich \u95ed\u4e0a\u94fe \u7684Lyapunov\u6307\u6570\u548c\u516c\u5f0f\u7684\u8bc1\u660e (\u8be5\u516c\u5f0f\u7531M.Kontsevich\u4e8e15\u5e74\u524d\u5728<a href=\"http:\/\/arxiv.org\/abs\/hep-th\/9701164\" target=\"_blank\">\u8fd9\u7bc7\u6587\u7ae0<\/a>\u4e2d\u63d0\u51fa)<br \/>\n \t<\/li><li>Alex Eskin and Maryam Mirzakhani\u6700\u8fd1\u5ba3\u79f0\u8bc1\u660e\u4e86Abel\u5fae\u5206\u6784\u6210\u7684\u6a21\u7a7a\u95f4(\u975e\u9f50\u6b21)\u4e2d\u7684\u4e00\u4e2a <a href=\"https:\/\/en.wikipedia.org\/wiki\/Ratner's_theorems\">Ratner\u578b\u5b9a\u7406<\/a>(\u4e0d\u53d8\u6d4b\u5ea6\u7684\u5206\u7c7b).<\/li><\/ol><!--more--><br \/>\n\u7279\u522b\u662f, \u9274\u4e8e\u6211\u5df2\u7ecf\u548cFernando Cod\u00e1 Marques, Andr\u00e9 Neves, Anton Zorich and Alex Eskin\u89c1\u8fc7\u51e0\u9762, \u5728\u6b64\u6211\u5f88\u9ad8\u5174(\u4e5f\u6709\u70b9\u5192\u5931\u5730)\u5bf9\u4ed6\u4eec\u6770\u51fa\u7684\u5de5\u4f5c\u8868\u793a\u8d5e\u7f8e! (\u4e8b\u5b9e\u4e0a, \u8be5\u6587\u9664\u4e86\u8bc1\u660e\u4e86\u4e00\u4e2a\u6f02\u4eae\u7684\u7ed3\u8bba, \u800c\u4e14\u4e5f\u662f\u7b80\u660e\u7684. \u4e0a\u8ff0\u5217\u8868\u4e2d\u7684\u7b2c\u4e00, \u7b2c\u4e09, \u7b2c\u56db\u5206\u522b\u670995, 106\u4ee5\u53ca152\u9875\u957f.:-)<\/p>\n<p>\u4e0b\u9762\u6211\u5c06\u5728\u672c\u535a\u5ba2\u4e2d\u5199\u4e00\u4e9b\u65e5\u5fd7\u6765\u5bf9\u524d\u9762\u63d0\u5230\u7684\u540e\u4e09\u9879\u4f5c\u4e9b\u6ce8\u8bb0. \u66f4\u786e\u5207\u7684\u8bf4, \u4eca\u5929\u7684\u8fd9\u7bc7\u65e5\u5fd7\u5c06\u5b8c\u5168\u7528\u6765\u8bf4\u660e\u7531Fernado \u548c Andr\u00e9 (\u8fd9\u91cc\u6211\u5c06\u7528\u540d\u5b57\u6765\u79f0\u547c\u4ed6\u4eec, \u56e0\u4e3a\u6211\u8ba4\u4e3a\u4ed6\u4eec\u4f1a\u4e0d\u4ecb\u610f) \u89e3\u51b3\u7684Willomroe\u731c\u60f3. \u800c\u540e, \u5c06\u5728\u53e6(\u4e00\u7bc7?)\u65e5\u5fd7\u4e2d\u8ba8\u8bbaAlex, Anton \u548c M. Kontsevich \u5173\u4e8eKontsevich-Zorich \u95ed\u4e0a\u94fe\u7684Lyapunov \u6307\u6570\u548c\u7684\u6587\u7ae0. \u6700\u540e, \u6211\u5c06\u5b8c\u6210\u8fd9\u4e00&#8221;\u7cfb\u5217&#8221;\u65e5\u5fd7, \u5e76\u7528(\u51e0\u7bc7?)\u65e5\u5fd7\u6765\u5b8c\u6210Alex \u548c M.Mirzakhani\u8bc1\u660e\u7684Ratner-\u578b\u5b9a\u7406, \u8fd9\u4e9b\u65e5\u5fd7\u5c06\u57fa\u4e8e\u524d\u51e0\u5468Alex\u5728Luminy\/Marseille\u6388\u8bfe\u7684\u7b14\u8bb0(\u7531\u6211\u8bb0\u5f55).<\/p>\n<p>\u5728\u8f6c\u5165\u6b63\u9898\u4e4b\u524d, \u5148\u8ba9\u6211\u4f5c\u4e00\u4e9b\u7533\u660e\u548c\u8bc4\u8bba. \u9996\u5148, \u5199\u4f5c\u4eca\u5929\u8fd9\u7bc7\u65e5\u5fd7\u4e4b\u6240\u4ee5\u88ab\u6781\u5927\u5730\u7b80\u5316\u4e86, \u662f\u7531\u4e8eFernado\u548cAndr\u00e9\u5728\u4ed6\u4eec\u6587\u7ae0\u7684\u7b2c\u4e8c\u8282\u5bf9\u5176\u8bc1\u660e\u601d\u60f3\u5199\u4e86\u4e00\u4e2a\u975e\u5e38\u6e05\u6670\u7684\u5927\u7eb2. \u7279\u522b\u662f, \u4eca\u5929\u6211\u5c06\u8ddf\u968f\u7740\u4ed6\u4eec\u7684\u5927\u7eb2, \u4f46\u540c\u65f6\u8bf7\u6ce8\u610f, \u6211\u53ef\u80fd\u5728\u8fd9\u4e2a\u8fc7\u7a0b\u4e2d\u72af\u4e00\u4e9b\u9519\u8bef (\u5f53\u7136, \u8fd9\u6837\u7684\u9519\u8bef\u5e94\u8be5\u7531\u6211\u4e00\u4eba\u627f\u62c5). \u7b2c\u4e8c, \u7b2c\u4e09\u6761\u7684\u6587\u7ae0\u5c06\u662fPascal Hubert\u4e8e2012\u5e7410\u6708\u5728\u5e03\u5c14\u5df4\u57fa\u7814\u8ba8\u73ed(S\u00e9minaire Bourbaki)\u7684\u62a5\u544a\u4e3b\u9898. \u7279\u522b\u662f, \u4ed6\u5c06\u4e3a\u6b64\u5199\u4e00\u4e2a\u4e25\u683c\u7684\u8bb2\u4e49, \u4e8e\u662f\u6211\u5173\u4e8e\u8fd9\u4e2a\u4e3b\u9898\u7684\u65e5\u5fd7\u5e94\u5f53\u88ab\u89c6\u4e3aPascal\u7b14\u8bb0\u7684\u975e\u6b63\u5f0f\u7248\u672c. \u6700\u540e, \u5173\u4e8e\u4e0a\u9762\u63d0\u5230\u7684\u7b2c\u56db\u6761, Alex Eskin\u5728\u4ed6\u7684\u4e3b\u9875(\u53c2\u8003<a href=\"http:\/\/www.math.uchicago.edu\/%7Eeskin\/luminy2012\/lectures.pdf\" target=\"_blank\">\u8fd9\u91cc<\/a>)\u6302\u4e86\u4e00\u4e2a\u975e\u5e38\u597d\u7684\u7b14\u8bb0(\u5173\u4e8e\u4ed6\u5f00\u8bbe\u7684\u5c0f\u8bfe\u7a0b(\u4e0d\u5230\u4e00\u4e2a\u5b66\u671f)\u7684). \u4e8b\u5b9e\u4e0a, Alex\u7684\u7b14\u8bb0\u8bfb\u8d77\u6765\u4e5f\u662f\u4ee4\u4eba\u6109\u60a6\u7684, \u4ece\u800c\u6211\u7acb\u9a6c\u6539\u53d8\u4e86\u4e00\u5f00\u59cb\u628a\u6211\u539f\u59cb\u7684\u7b14\u8bb0\u5199\u5230\u8fd9\u91cc\u7684\u521d\u8877, \u8fd9\u6837\u6211\u5c06\u505a\u4e9bAlex\u548cM. Mirzakhani\u7528\u5230\u7684\u6280\u5de7\u7684\u8bc4\u8bba, \u4f8b\u5982, \u6211\u5c06\u8bd5\u7740\u8d34\u51fa\u6211\u4eceJ.-F. Quint\u5f00\u8bbe\u7684\u5c0f\u8bfe\u7a0b(\u4e5f\u8bf7\u53c2\u8003 Luminy\/Marseille)\u8bb0\u7684\u7b14\u8bb0\u4e2d\u6240\u8c13\u7684&#8221;\u6307\u6570\u6f02\u79fb&#8221;(exponential drift)\u60f3\u6cd5(\u6765\u6e90\u4e8e\u4ed6\u548cY. Benoist\u7684<a href=\"http:\/\/yunpan.cn\/lk\/19qt3wfrvc\" target=\"_blank\">\u8457\u540d\u6587\u7ae0<\/a>).<\/p>\n<p>\u5728\u8bf4\u5b8c\u8fd9\u4e9b\u8bc4\u8bba\u4e4b\u540e, \u8ba9\u6211\u4eec(\u4ece\u4e0b\u6bb5)\u5f00\u59cb\u5bf9Willmore\u731c\u60f3\u4ee5\u53caFernado\u548cAndr\u00e9\u5bf9\u5176\u89e3\u51b3\u7684\u8ba8\u8bba.<br \/>\n<span id=\"contents\"  style=\"text-align:center; font-size:18px; font-variant:small-caps;display:block;\">\u76ee\u5f55<\/span><br \/>\n          <span id=\"sec:content\"><a href=\"#contents\">\u76ee\u5f55<\/a><\/span><br \/><span>&#x00A0;1.&#x00A0;&#x00A0;<a href=\"#sec:1\">\u5f15\u8a00<\/a><\/span><br \/><span>&#x00A0;2.&#x00A0;&#x00A0;<a href=\"#sec:2\">\u5b9a\u74063\u7684\u8bc1\u660e\u7b56\u7565<\/a><\/span><br \/><span>&#x00A0;&#x00A0;&#x00A0;2.1.&#x00A0;&#x00A0;<a href=\"#sec:2.1\">\u6781\u5c0f&#8212;\u6781\u5927\u7406\u8bba\u7684\u5178\u578b\u4f8b\u5b50<\/a><\/span><br \/><span>&#x00A0;&#x00A0;&#x00A0;2.2.&#x00A0;&#x00A0;<a href=\"#sec:2.2\">\u5178\u5219\u66f2\u9762\u7c07<\/a><\/span><br \/><span>&#x00A0;&#x00A0;&#x00A0;2.3.&#x00A0;&#x00A0;<a href=\"#sec:2.3\">\u7206\u7834\u5904\u7406<\/a><\/span><br \/><span>&#x00A0;&#x00A0;&#x00A0;2.4.&#x00A0;&#x00A0;<a href=\"#sec:2.4\">$\\Sigma\\subset S^3$\u7684\u6781\u5c0f\u6781\u5927\u540c\u4f26\u7c07<\/a><\/span><br \/><br \/>\n<span class=\"latex_section\">1.&#x00A0;\u5f15\u8a00<a id=\"sec:1\"><\/a><\/span>\n\n\u5047\u8bbe $\\widetilde{\\Sigma}\\hookrightarrow\\mathbb{R}^3$ \u662f ${\\mathbb{R}^3}$ \u4e00\u4e2a\u95ed\u7684(\u5373\u7d27\u81f4\u4e14\u65e0\u8fb9\u754c)\u6d78\u5165\u66f2\u9762. \u5728 ${\\widetilde{\\Sigma}}$ \u7684\u6240\u6709\u6bd4\u8f83\u7b80\u5355\u7684\u51e0\u4f55\u4e0d\u53d8\u91cf\u4e2d, \u6211\u4eec\u53ef\u4ee5\u9009\u53d6(\u6bcf\u70b9 ${p\\in\\widetilde{\\Sigma}}$ \u7684)\u9ad8\u65af\u66f2\u7387 ${\\widetilde{K}=\\widetilde{K}(p)}$ \u4ee5\u53ca\u5e73\u5747\u66f2\u7387 ${\\widetilde{H}=\\widetilde{H}(p)}$, \u4ed6\u4eec\u662f\u6240\u8c13\u7684\u7b2c\u4e8c\u57fa\u672c\u578b ${\\widetilde{A}=\\widetilde{A}(p):T_p\\widetilde{\\Sigma}\\rightarrow T_p\\widetilde{\\Sigma}}$ \u7684\u884c\u5217\u5f0f(determinant)\u548c\u8ff9(trace) (\u4e00\u4e2a ${2\\times2}$ \u7684\u7ebf\u6027\u53d8\u6362\/\u77e9\u9635\u7684\u7279\u5f81\u503c\u662f ${\\widetilde{\\Sigma}}$ \u5728 $p$ \u70b9\u7684\u4e3b\u66f2\u7387(principal curvatures)). \u5173\u4e8e\u8fd9\u4e9b\u6982\u5ff5\u7684\u9610\u8ff0, \u4e00\u672c\u975e\u5e38\u597d\u7684\u53c2\u8003\u4e66\u662f M. do Carmo \u7684\u7ecf\u5178\u8457\u4f5c: \u66f2\u7ebf\u4e0e\u66f2\u9762\u7684\u5fae\u5206\u51e0\u4f55(Differential geometry of curves and surfaces).<br \/>\n\u7528\u8fd9\u79cd(\u5fae\u5206\u51e0\u4f55\u7684)\u8bed\u8a00, Willmore \u80fd\u91cf(Willmore Energy) ${\\widetilde{\\mathcal{W}}(\\widetilde{\\Sigma})}$ \u5b9a\u4e49\u4e3a<br \/>\n\\[<br \/>\n\\widetilde{\\mathcal{W}}(\\widetilde{\\Sigma}) = \\int_{\\widetilde{\\Sigma}} \\widetilde{H}^2\\,d\\widetilde{\\Sigma}<br \/>\n\\]<br \/>\n\u5176\u4e2d ${d\\widetilde{\\Sigma}}$ \u662f ${\\widetilde{\\Sigma}}$ \u7684\u9762\u79ef\u5143.<br \/>\nWillmore \u80fd\u91cf\u81ea\u7136\u7684\u51fa\u73b0\u5728\u67d0\u4e9b\u7269\u7406(\u5728\u5f39\u6027\u58f3(elastic shells))\u548c\u751f\u7269(\u7ec6\u80de\u819c(cell membranes))\u7684\u7814\u7a76\u4e2d, \u5728\u90a3\u91cc, \u6709\u65f6\u4e5f\u88ab\u79f0\u4e3a\u626d\u66f2\u80fd\u91cf(bending energy).<br \/>\n\u5728\u6570\u5b66\u4e2d, \u6211\u4eec\u77e5\u9053\u5b83\u5728 ${\\mathbb{R}^3}$\u4e2d\u7684\u5171\u5f62\u53d8\u6362\u4e0b\u662f\u4e0d\u53d8\u7684(\u53c2\u8003 W. Blaschke), \u800c\u4e14\u5b83\u6709\u4e00\u4e2a\u975e\u5e38\u81ea\u7136\u7684\u51e0\u4f55(\u53d8\u5206)\u7ed3\u6784,\u5373\u5728\u6240\u6709\u6d78\u5165\u95ed\u66f2\u9762 ${\\widetilde{\\Sigma}\\hookrightarrow\\mathbb{R}^{3}}$ \u4e2d\u4ed6\u4f55\u65f6\u8fbe\u5230\u6700\u5c0f?<br \/>\n\u8fd9\u91cc, \u53ef\u4ee5\u8bc1\u660e\u5bf9\u6240\u6709\u6d78\u5165\u95ed\u66f2\u9762 ${\\widetilde{\\Sigma}\\hookrightarrow\\mathbb{R}^3}$, \u6709<br \/>\n\\[<br \/>\n\\widetilde{\\mathcal{W}}(\\widetilde{\\Sigma})\\geq 4\\pi = \\widetilde{\\mathcal{W}}(S^2(r))<br \/>\n\\]<br \/>\n\u5176\u4e2d, ${S^2(r)\\subset\\mathbb{R}^3}$ \u662f\u534a\u5f84\u4e3a ${r&gt;0}$ \u7684 (??\u6807\u51c6\u7403\u9762, \u539f\u6587\u8fd9\u91cc\u7f3a\u5931). \u800c\u4e14 ${\\widetilde{\\mathcal{W}}(\\widetilde{\\Sigma})= 4\\pi}$ \u5f53\u4e14\u4ec5\u5f53 ${\\widetilde{\\Sigma}}$ \u662f\u4e00\u4e2a\u7403\u9762(\u6807\u51c6\u7403\u9762, \u975e\u62d3\u6251\u7403\u9762). \u6362\u8a00\u4e4b, \u5728 ${\\mathbb{R}^3}$ \u4e2d\u7684\u6807\u51c6\u7403\u662f\u5728\u6240\u6709\u6d78\u5165\u95ed\u66f2\u9762\u4e2d\u4f7fWillmore\u80fd\u91cf\u6781\u5c0f\u8005. \u4e8b\u5b9e\u4e0a, \u8fd9\u4e00\u4e8b\u5b9e\u7684\u8bc1\u660e\u5e76\u4e0d\u56f0\u96be:<br \/>\n<ul><li>\u8bb0 ${k_1\\geq k_2}$ \u4e3a\u66f2\u9762\u7684\u4e3b\u66f2\u7387, \u56de\u5fc6\u5e73\u5747\u66f2\u7387\u662f${\\widetilde{H}=(k_1+k_2)\/2}$, \u800c\u9ad8\u65af\u66f2\u7387\u662f ${\\widetilde{K}=k_1k_2}$. \u5229\u7528\u6052\u7b49\u5f0f ${(k_1+k_2)^2=(k_1-k_2)^2+4k_1k_2}$, \u6709<br \/>\n\\[<br \/>\n\\widetilde{\\mathcal{W}}(\\widetilde{\\Sigma})=\\int_{\\widetilde{\\Sigma}} \\widetilde{H}^2\\,d\\widetilde{\\Sigma} = \\frac{1}{4}\\int_{\\widetilde{\\Sigma}} (k_1-k_2)^2\\,d\\widetilde{\\Sigma}+\\int_{\\widetilde{\\Sigma}} \\widetilde{K}\\,d\\widetilde{\\Sigma}\\geq \\int_{\\widetilde{\\Sigma}} \\widetilde{K}\\,d\\widetilde{\\Sigma}<br \/>\n\\]<br \/>\n\u53e6\u4e00\u65b9\u9762, \u7531\u9ad8\u65af-\u535a\u5185\u5b9a\u7406(Gauss-Bonnet theorem), ${\\int_{\\widetilde{\\Sigma}} \\widetilde{K}\\,d\\widetilde{\\Sigma}=2\\pi\\chi(\\widetilde{\\Sigma})}$. \u56e0\u6b64, \u5047\u8bbe ${\\widetilde{\\Sigma}}$ \u662f\u62d3\u6251\u7403\u9762, \u5373\u5b83\u7684\u4e8f\u683c $g=0$, \u4ece\u800c\u6b27\u62c9\u793a\u6027\u6570\u4e3a ${\\chi(\\widetilde{\\Sigma})=2-2g=2}$, \u6211\u4eec\u53ef\u4ee5\u5f97\u5230<br \/>\n\\[<br \/>\n\\widetilde{\\mathcal{W}}(\\widetilde{\\Sigma})\\geq \\int_{\\widetilde{\\Sigma}} \\widetilde{K}\\,d\\widetilde{\\Sigma}=4\\pi<br \/>\n\\]<br \/>\n\u800c\u4e14\u7b49\u53f7\u6210\u7acb\u5f53\u4e14\u4ec5\u5f53<br \/>\n\\[<br \/>\n\\frac{1}{4}\\int_{\\widetilde{\\Sigma}} (k_1-k_2)^2\\,d\\widetilde{\\Sigma}=0<br \/>\n\\]<br \/>\n\u5373 ${k_1=k_2}$ \u5bf9 ${\\widetilde{\\Sigma}}$ \u4e0a\u6240\u6709\u70b9\u90fd\u6210\u7acb. \u5373, \u6240\u6709\u70b9\u90fd\u662f\u5e73\u70b9, \u8fd9\u6837\u5c31\u8bf4\u660e ${\\widetilde{\\mathcal{W}}(\\widetilde{\\Sigma})=4\\pi}$ \u6210\u7acb\u5f53\u4e14\u4ec5\u5f53 ${\\widetilde{\\Sigma}}$ \u662f\u4e2a\u6807\u51c6\u7403.<\/li><li>\u4e00\u822c\u5730(\u5373, \u5f53 ${\\widetilde{\\Sigma}}$ \u7684\u4e8f\u683c $g\\geq1$ \u65f6), \u524d\u9762\u7684\u8ba1\u7b97\u4e0d\u518d\u6709\u6548, \u56e0\u4e3a ${\\int_{\\widetilde{\\Sigma}} \\widetilde{K}\\,d\\widetilde{\\Sigma}=2\\pi\\chi(\\widetilde{\\Sigma})=2\\pi(2-2g)\\leq 0}$, \u5373\u4f9d\u8d56\u4e8e\u9ad8\u65af-\u535a\u5185\u5b9a\u7406\u800c\u5f97\u5230 ${\\widetilde{\\mathcal{W}}(\\widetilde{\\Sigma})\\geq \\int_{\\widetilde{\\Sigma}} \\widetilde{K}\\,d\\widetilde{\\Sigma}}$ \u7684\u4f30\u8ba1, \u6bd4\u5e73\u51e1\u7684\u754c ${\\widetilde{\\mathcal{W}}(\\widetilde{\\Sigma})= \\int_{\\widetilde{\\Sigma}} H^2\\,d\\widetilde{\\Sigma}\\geq0}$ \u8981\u5f31. \u7136\u800c\u6211\u4eec\u53ef\u4ee5\u53d6\u800c\u4ee3\u4e4b\u7528 Chern-Lashof \u4e0d\u7b49\u5f0f<br \/>\n\\[<br \/>\n\\int_{\\widetilde{\\Sigma}} |\\widetilde{K}|\\,d\\widetilde{\\Sigma}\\geq 2\\pi(2+2g)<br \/>\n\\]<br \/>\n\u6765\u514b\u670d\u8fd9\u70b9: \u4e8b\u5b9e\u4e0a, \u82e5\u8bb0 ${a:=\\int_{\\{\\widetilde{K}&gt;0\\}} \\widetilde{K}\\,d\\widetilde{\\Sigma}}$ \u548c ${b:=\\int_{\\{\\widetilde{K}\\leq0\\}} \\widetilde{K}\\,d\\widetilde{\\Sigma}}$, \u5219\u6211\u4eec\u6709<br \/>\n\\[a+b=\\int_{\\widetilde{\\Sigma}} \\widetilde{K}\\,d\\widetilde{\\Sigma}=(2-2g)2\\pi<br \/>\n\\]<br \/>\n\u4ee5\u53ca<br \/>\n\\[<br \/>\na-b=\\int_{\\widetilde{\\Sigma}} |\\widetilde{K}|\\,d\\widetilde{\\Sigma}\\geq 2\\pi(2+2g)<br \/>\n\\]<br \/>\n\u4e8e\u662f ${a\\geq 4\\pi}$. \u73b0\u5728, \u6211\u4eec\u5bf9\u533a\u57df ${\\{\\widetilde{K}&gt;0\\}}$ \u53ef\u4ee5\u7528 ${(k_1+k_2)^2=(k_1-k_2)^2+4k_1k_2}$ \u5f97\u5230<br \/>\n\\[<br \/>\n4\\pi\\leq a\\leq \\int_{\\{\\widetilde{K}&gt;0\\}} \\widetilde{K}\\,d\\widetilde{\\Sigma} + \\frac{1}{4}\\int_{\\{\\widetilde{K}&gt;0\\}} (k_1-k_2)^2\\,d\\widetilde{\\Sigma}\\leq \\widetilde{\\mathcal{W}}(\\widetilde{\\Sigma})<br \/>\n\\]<\/li><\/ul>\u65e0\u8bba\u5bf9\u8fd9\u4e24\u79cd\u60c5\u51b5\u4e2d\u7684\u54ea\u4e00\u79cd, \u4e00\u4f46\u610f\u8bc6\u5230\u5728\u6240\u6709\u6d78\u5165${\\mathbb{R}^3}$\u7684\u7d27\u66f2\u9762\u4e2d\u6781\u5c0f\u5316Willmore\u80fd\u91cf\u5e76\u4e0d\u56f0\u96be, \u90a3\u4e48\u6211\u4eec\u4e5f\u8bb8\u4f1a\u548cT. Willmore\u4e00\u6837\u8003\u8651\u5728\u6240\u6709\u6d78\u5165${\\mathbb{R}^3}$\u7684\u73af\u9762\u4e2d\u4f7fWillmore\u80fd\u91cf\u6700\u5c0f\u8fd9\u4e2a\u95ee\u9898(\u6700\u5c0f\u662f\u591a\u5c11? \u4ec0\u4e48\u65f6\u5019\u8fbe\u5230\u8fd9\u4e2a\u6700\u5c0f?).<br \/>\n<span id='thm:willmore_conj'><\/span><div class='latex_thm'><span class='latex_thm_h'>\u5b9a\u7406 1<\/span> (<span class='latex_thm_name'>Willmore \u731c\u60f3, (1965)<\/span>)<span class='latex_thm_h'>.<\/span> \u5bf9\u6240\u6709\u7684\u6d78\u5165\u73af\u9762${\\widetilde{\\Sigma}\\hookrightarrow\\mathbb{R}^3}$, \u90fd\u6709<br \/>\n\\[{\\widetilde{W}(\\widetilde{\\Sigma})\\geq 2\\pi^2(&gt;4\\pi)}.\\]<br \/>\n\u5bf9\u65cb\u8f6c\u73af\u9762 ${\\widetilde{\\Sigma}_c}$ \u7b49\u5f0f ${\\widetilde{W}(\\widetilde{\\Sigma}_c)=2\\pi^2}$ \u6210\u7acb, \u8fd9\u4e2a\u73af\u9762\u7531\u4e00\u4e2a\u534a\u5f84\u4e3a $1$, \u5176\u4e2d\u5fc3\u79bb\u8f74\u5411(z\u8f74)\u8ddd\u79bb\u4e3a ${\\sqrt{2}}$ \u7684\u5706\u7ed5 z\u8f74 \u65cb\u8f6c\u800c\u5f97. \u5176\u65b9\u7a0b\u53ef\u4ee5\u5199\u6210:<br \/>\n\\[<br \/>\n(u,v)\\mapsto ((\\sqrt{2}+\\cos u)\\cos v, (\\sqrt{2}+\\cos u) \\sin v, \\sin u)\\in\\mathbb{R}^3.<br \/>\n\\]<br \/>\n<\/div><br \/>\n\u5173\u4e8e\u7b49\u53f7\u53d6\u5f97\u65f6\u7684\u8fd9\u4e2a\u73af\u9762, \u53ef\u4ee5\u53c2\u8003<a title=\"clifford torus\" href=\"http:\/\/nylander.wordpress.com\/2008\/08\/12\/clifford-torus\/\" target=\"_blank\">P.Nylander\u7684\u4e00\u7bc7\u65e5\u5fd7<\/a>, \u90a3\u91cc\u6709\u5b83\u7684\u4e00\u4e2a\u51c6\u786e\u56fe\u5f62.<\/p>\n<figure style=\"width: 781px\" class=\"wp-caption aligncenter\"><a href=\"http:\/\/i.imgur.com\/cxzHy.png\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/i.imgur.com\/cxzHy.png\" width=\"781\" height=\"714\" alt=\"Hopf Fiberation\" class=\"size-medium\" \/><\/a><figcaption class=\"wp-caption-text\">Hopf Fiberation<\/figcaption><\/figure>\n<p>\u8fd9\u4e2a\u65cb\u8f6c\u73af\u9762${\\widetilde{\\Sigma}_c}$\u662fClifford\u73af\u9762${\\Sigma_c=S^1(1\/\\sqrt{2})\\times S^1(1\/\\sqrt{2})\\subset S^3}$\u5728\u7403\u6781\u6295\u5f71${\\pi:S^3-\\{(0,0,0,1)\\}\\rightarrow\\mathbb{R}^3}$\u4e0b\u7684\u50cf, \u5373, ${\\widetilde{\\Sigma}_c=\\pi(\\Sigma_c)}$.<\/p>\n<p>\u66f4\u4e00\u822c\u5730, Willmore\u731c\u60f3\u53ef\u4ee5\u89c6\u4e3a\u5173\u4e8e\u901a\u8fc7\u7403\u6781\u6295\u5f71$\\pi$\u800c\u6d78\u5165(immersed)\u5230$S^3$\u7684\u95ed\u66f2\u9762\u7684\u4e00\u4e2a\u95ee\u9898. \u4e8b\u5b9e\u4e0a, \u7ed9\u5b9a${\\Sigma\\subset S^3}$, \u901a\u8fc7\u4ee4 ${\\widetilde{\\Sigma}=\\pi(\\Sigma)}$, \u6211\u4eec\u6709<br \/>\n\\[<br \/>\n\\widetilde{W}(\\widetilde{\\Sigma}) = \\int_{\\widetilde{\\Sigma}} \\widetilde{H}^2\\,d\\widetilde{\\Sigma} = \\int_{\\Sigma}(1+H^2)\\,d\\Sigma:=\\mathcal{W}(\\Sigma)<br \/>\n\\]<br \/>\n\u5176\u4e2d, $H$ \u662f${\\Sigma\\subset S^3}$\u7684\u5e73\u5747\u66f2\u7387. \u56e0\u6b64, \u6781\u5c0f\u5316${\\mathcal{W}(\\widetilde{\\Sigma})}$\u7b49\u4ef7\u4e8e\u6781\u5c0f\u5316${\\mathcal{W}(\\Sigma)}$. \u6b63\u56e0\u4e3a\u5982\u6b64, \u6211\u4eec\u4e5f\u5c06${\\mathcal{W}(\\Sigma)}$\u79f0\u4e3a${\\Sigma\\subset S^3}$\u7684Willmore\u80fd\u91cf(\u6cdb\u51fd).<\/p>\n<p>\u5c3d\u7ba1\u8fd9\u4e2a\u770b\u6cd5\u770b\u4f3c\u5f88\u5e73\u51e1, \u4f46\u662f\u4ed6\u5374\u8868\u660e\u4e86\u4e00\u4e2a\u6781\u6709\u610f\u4e49\u7684\u4e8b\u5b9e:Willmore\u731c\u60f3\u662f\u4e00\u4e2a\u53d8\u5206(\u6781\u5c0f\u5316)\u95ee\u9898. \u4f8b\u5982, \u6211\u4eec\u8003\u8651\u4e00\u5217\u66f2\u9762$M_i$\u6536\u655b\u4e8eWillmore\u80fd\u91cf\u7684\u4e0b\u786e\u754c, \u5e76\u901a\u8fc7\u6cbf\u7740\u8fd9\u5217$M_i$\u53d6\u67d0\u79cd&#8221;\u6781\u9650&#8221;\u800c\u6784\u9020\u4e00\u4e2a\u66f2\u9762$M$\u6781\u5c0f\u5316Willmore\u80fd\u91cf, \u90a3\u4e48${M_i\\subset S^3}$\u5c31\u6bd4${M_i\\subset\\mathbb{R}^3}$\u597d, \u56e0\u4e3a $S^3$\u662f\u7d27\u81f4\u7684, \u800c${\\mathbb{R}^3}$\u5e76\u4e0d\u662f\u7d27\u81f4\u7684.<\/p>\n<p>\u5173\u4e8eWillmore\u731c\u60f3, \u6211\u4eec\u5df2\u7ecf\u6709\u4e9b\u5df2\u77e5\u7684\u7ed3\u679c:<br \/>\n<ol><li>P.Li \u548c S.-T. Yau[<a href='#LiYau1982new'>4<\/a>] \u8bc1\u660e\u5982\u679c\u4e00\u4e2a\u6d78\u5165${f:\\Sigma\\rightarrow S^3}$ \u8986\u76d6\u67d0\u4e2a\u70b9 ${x\\in S^3}$ \u81f3\u5c11 ${k\\geq 1}$ \u6b21, \u90a3\u4e48${\\mathcal{W}(\\Sigma)\\geq 4\\pi k}$. \u7279\u522b\u5730, \u5982\u679c$\\Sigma$\u662f\u4e00\u4e2a\u6d78\u5165\u800c\u975e\u5d4c\u5165\u66f2\u9762(\u5373 $k &gt; 1$ \u5728\u67d0\u70b9$x\\in S^3$), \u90a3\u4e48 ${\\mathcal{W}(\\Sigma)\\geq 8\\pi&gt;2\\pi^2}$. \u6362\u8a00\u4e4b, \u6211\u4eec\u53ea\u9700\u8981\u5bf9\u5d4c\u5165\u66f2\u9762${\\Sigma\\hookrightarrow S^3}$\u4f30\u8ba1${\\mathcal{W}(\\Sigma)}$;<\/li><li>L. Simon[<a href='#Simon1993Existence'>9<\/a>] \u8bc1\u660e\u5b58\u5728\u4e00\u4e2a\u73af\u9762\u6781\u5c0f\u5316Willmore\u80fd\u91cf\u6cdb\u51fd(\u800c\u540e, \u8fd9\u4e2a\u7ed3\u679c\u88abM. Bauer\u548cE.Kuwert[<a href='#MatthiasErnst2003Existence'>1<\/a>]\u63a8\u5e7f\u5230\u9ad8\u4e8f\u683c\u60c5\u5f62);<\/li><li>Willmore\u80fd\u91cf\u6cdb\u51fd${\\mathcal{W}}$\u7684\u4e34\u754c\u70b9$\\Sigma$\u79f0\u4e3aWillmore\u66f2\u9762, \u8fd9\u4e2a\u6cdb\u51fd\u7684Euler-Lagrange\u65b9\u7a0b\u662f<br \/>\n\\[<br \/>\n\\Delta H + 2(H^2-K)H=0<br \/>\n\\]<br \/>\n\u5176\u4e2d$\\Delta$\u662f$\\Sigma$\u4e0a\u7684\u62c9\u666e\u62c9\u65af(Laplacian), $K$\u662f\u66f2\u9762$\\Sigma$\u7684\u9ad8\u65af\u66f2\u7387. \u4ece\u8fd9\u4e2a\u65b9\u7a0b(\u6709\u65f6\u79f0\u4e4b\u4e3aWillmore\u65b9\u7a0b), \u6211\u4eec\u770b\u5230$S^3$\u4e2d\u7684\u6781\u5c0f\u66f2\u9762(\u53c2\u8003B. Lawson[<a href='#Lawson1970Complete'>3<\/a>]\u5728\u8fd9\u7bc7\u6587\u7ae0\u4e2d\u6784\u9020\u7684\u4f8b\u5b50), \u5373${H\\equiv 0}$, \u662fWillmore\u66f2\u9762, \u4f46\u662f\u5e76\u4e0d\u662f\u6240\u6709\u7684Willmore\u66f2\u9762\u90fd\u662f\u5982\u6b64(\u53c2\u8003R. Bryant[<a href='#Bryant1984duality'>2<\/a>], U.Pinkall[<a href='#Pinkall1985Hopf'>6<\/a>]\u7684\u6587\u7ae0). \u5173\u4e8eWillmore\u65b9\u7a0b\u5206\u6790\u65b9\u9762\u66f4\u591a\u7684\u4fe1\u606f, \u8bf7\u53c2\u8003T. Rivi\u00e8re \u7684\u6587\u7ae0[<a href='#Riviere2008Willmore'>7<\/a>].<\/li><\/ol>\u5728\u6700\u8fd1\u4e00\u7bc7\u7a81\u7834\u6027\u7684\u6587\u7ae0\u4e2d, Fernando Cod\u00e1 Marques \u548c Andr\u00e9 Neves \u8bc1\u660e\u4e86\u5982\u4e0b\u5b9a\u7406:<br \/>\n<span id='thm:main'><\/span><div class='latex_thm'><span class='latex_thm_h'>\u5b9a\u7406 2<\/span> (<span class='latex_thm_name'>F. C. Marques and A. Neves<\/span>)<span class='latex_thm_h'>.<\/span> \u5047\u8bbe$ {\\Sigma\\hookrightarrow S^3} $\u662f\u4e00\u4e2a\u4e8f\u683c${g\\geq 1}$\u7684\u5d4c\u5165\u95ed\u66f2\u9762, \u90a3\u4e48<br \/>\n\\[<br \/>\n\\mathcal{W}(\\Sigma)\\geq 2\\pi^2.<br \/>\n\\]<br \/>\n\u800c\u4e14, ${\\mathcal{W}(\\Sigma)=2\\pi^2}$\u5f53\u4e14\u4ec5\u5f53$\\Sigma$\u548cClifford\u73af\u9762 ${\\Sigma_c}$\u5171\u5f62.<br \/>\n<\/div><br \/>\n\u7531\u4e0a\u9762Li-Yau\u7684\u7ed3\u679c, \u8fd9\u4e2a\u6df1\u523b\u7684\u7ed3\u679c\u8574\u542bWIllmore\u731c\u60f3\u6210\u7acb.<\/p>\n<p>\u5173\u4e8e<a class='latex_ref' href=#thm:main>\u5b9a\u7406 2<\/a>\u7684\u8bc1\u660e, Fernando \u548c Andr\u00e9 \u9996\u5148\u5229\u7528\u6781\u5c0f-\u6781\u5927\u5b9a\u7406(\u53c2\u8003\u4e0b\u9762\u7684<a class='latex_ref' href=#thm:min-max>\u5b9a\u7406 3<\/a>)\u628a\u4e00\u822c\u60c5\u5f62\u4e0b\u7684$\\Sigma$\u5316\u5f52\u4e3a\u6781\u5c0f\u66f2\u9762${\\Sigma}$.<br \/>\n<div class='latex_rem'><span class='latex_rem_h'>\u6ce8\u8bb0 1<\/span><span class='latex_rem_h'>.<\/span> \u8fd9\u91cc, \u5728$0$-\u6b21\u903c\u8fd1\u4e2d, \u8fd9\u4e2a\u5316\u5f52\u80cc\u540e\u7684&#8221;\u539f\u7406&#8221;\u662fWillmore\u80fd\u91cf<br \/>\n\\[<br \/>\n\\mathcal{W}(\\Sigma)=\\int_{\\Sigma}(1+H^2)\\,d\\Sigma = \\int_{\\Sigma}1 \\,d\\Sigma=\\textrm{area}(\\Sigma)<br \/>\n\\]<br \/>\n\u5728$\\Sigma$\u662f\u6781\u5c0f\u66f2\u9762\u65f6\u4e0e\u9762\u79ef\u6cdb\u51fd\u4e00\u81f4, \u800c\u6781\u5c0f\u66f2\u9762\u7684\u9762\u79ef\u6cdb\u51fd\u7684\u7814\u7a76\u662f\u5fae\u5206\u51e0\u4f55\u4e2d\u4e00\u4e2a\u7ecf\u5178\u7684\u8bfe\u9898.<br \/>\n<\/div><br \/>\n\u7c97\u7565\u6765\u8bf4, \u7ed9\u5b9a\u4e00\u4e2a\u4e00\u822c\u7684\u5d4c\u5165\u95ed\u66f2\u9762${\\Sigma\\subset S^3}$, \u5176\u4e8f\u683c${g\\geq 1}$, \u4ed6\u4eec\u8003\u8651\u5b83\u7684&#8221;\u6781\u5c0f-\u6781\u5927\u540c\u4f26\u7c7b&#8221;(min-max homotopy class)$\\pi$\u4ee5\u53ca\u5bf9\u5e94\u7684\u5bbd\u5ea6(width)$L(\\pi)$. \u975e\u6b63\u5f0f\u5730\u8bf4, \u5bf9\u7d27\u81f43\u7ef4\u6d41\u5f62$M^3$\u4e2d\u4e00\u4e2a2-\u7ef4\u66f2\u9762, \u6211\u4eec\u53ef\u4ee5\u63cf\u8ff0\u5b83\u7684\u4e00\u822c\u540c\u4f26\u7c7b(general homotopy classes)\u4ee5\u53ca\u5b83\u4eec\u7684\u5bbd\u5ea6\u5982\u4e0b:<br \/>\n\u7ed9\u5b9a${n\\geq 1}$, \u4ee4${I^n=[0,1]^n} $\u662f${\\mathbb{R}^n}$\u4e2d\u7684\u5355\u4f4d\u65b9\u4f53, \u4ee4$\\Phi$\u662f\u4e00\u4e2a\u5b9a\u4e49\u5728$I^n$\u4e0a\u7684(\u8fde\u7eed)\u6620\u5c04, \u4f7f\u5f97\u5bf9\u6bcf\u70b9$x\\in I^n$, ${\\Phi(x)}$\u662f$M^3$\u4e2d\u4e00\u4e2a\u7d27\u81f4\u65e0\u8fb9\u66f2\u9762. \u6211\u4eec\u79f0\u4e24\u4e2a\u8fd9\u6837\u7684\u6620\u5c04$\\Phi_0$\u548c$\\Phi_1$\u662f\u540c\u4f26\u7684(\u76f8\u5bf9\u4e8e$\\partial I^n$)\u5982\u679c\u5b58\u5728\u4e00\u4e2a\u5b9a\u4e49\u5728$I^{n+1}=I\\times I^n$\u4e0a\u7684\u8fde\u7eed\u6620\u5c04$\\Psi$, \u4f7f\u5f97<\/p>\n<ol>\n<li>\u5bf9\u6240\u6709\u7684${y=(t,x)\\in I^{n+1}=I\\times I^n}$, ${\\Psi(y)}$\u662f$M^3$\u7684\u4e00\u4e2a\u7d27\u81f4\u65e0\u8fb9\u66f2\u9762;<\/li>\n<li>\u5bf9\u6240\u6709\u7684$x\\in I^n$\u6709 ${\\Psi(0,x)=\\Phi_0(x)}$\u4ee5\u53ca${\\Psi(1,x)=\\Phi_1(x)}$;<\/li>\n<li>\u5bf9\u6240\u6709\u7684${t\\in I}, {x\\in\\partial I^n}$, \u6709${\\Psi(t,x)=\\Phi_0(x)=\\Phi_1(x)}$.<\/li>\n<\/ol>\n<p>\u5728\u5982\u4e0b\u7684\u4e24\u4e2a\u56fe\u4e2d, \u6211\u4eec\u6f14\u793a\u4e86\u4f4e\u7ef4\u60c5\u5f62\u4e0b\u540c\u4f26\u7684\u6982\u5ff5, \u5373 $M=S^2$(\u800c\u4e0d\u662f\u4e00\u4e2a\u4e09\u7ef4\u6d41\u5f62)\u4ee5\u53ca\u6620\u5c04$\\Phi$\u53d6\u503c\u4e8e\u66f2\u7ebf(\u800c\u4e0d\u662f\u66f2\u9762). \u5728\u7b2c\u4e00\u4e2a\u56fe\u4e2d, \u6211\u4eec\u53d6$n=0$, \u56e0\u6b64 $I^0=\\set{0}$, \u6211\u4eec\u5206\u522b\u7528\u7eff\u7ebf\u548c\u7ea2\u7ebf\u8868\u793a$I^0$\u5728\u6620\u5c04$\\Phi_0$\u548c$\\Phi_1$\u4e0b\u7684\u50cf. \u800c\u5728\u7b2c\u4e8c\u4e2a\u56fe\u4e2d, \u6211\u4eec\u4ee4$n=1$, \u4e8e\u662f$I^1=I$, \u540c\u6837$\\Phi_0$\u548c$\\Phi_1$\u7684\u50cf\u7528\u7eff\u8272\u548c\u7ea2\u8272\u533a\u5206. \u800c$\\partial I$\u7684\u8fb9\u754c\u662f\u4e24\u6761\u5e73\u51e1(\u79bb\u6563)&#8221;\u66f2\u7ebf&#8221;, \u5bf9\u5e94\u4e8e$S^2$\u7684\u5317\u6781\u70b9\u548c\u5357\u6781\u70b9.<\/p>\n<figure style=\"width: 280px\" class=\"wp-caption aligncenter\"><a href=\"http:\/\/i.imgur.com\/x1vPc.jpg\"><img loading=\"lazy\" decoding=\"async\" title=\"\u4ece $I^0=\\set{0}$\u5230$S^2$\u4e0a\u66f2\u7ebf\u7684\u4e24\u4e2a\u6620\u5c04$\\Phi_0$\u548c$\\Phi_1$\" src=\"http:\/\/i.imgur.com\/x1vPc.jpg\" alt=\"\u4ece $I^0=\\set{0}$\u5230$S^2$\u4e0a\u66f2\u7ebf\u7684\u4e24\u4e2a\u6620\u5c04$\\Phi_0$\u548c$\\Phi_1$\" width=\"280\" height=\"224\" \/><\/a><figcaption class=\"wp-caption-text\">\u4ece $I^0=\\set{0}$\u5230$S^2$\u4e0a\u66f2\u7ebf\u7684\u4e24\u4e2a\u6620\u5c04${\\Phi_0}$\u548c${\\Phi_1}$<\/figcaption><\/figure>\n<figure style=\"width: 280px\" class=\"wp-caption aligncenter\"><a href=\"http:\/\/i.imgur.com\/C9cpt.jpg\"><img loading=\"lazy\" decoding=\"async\" title=\"\u4ece $I^1=I$\u5230$S^2$\u4e0a\u66f2\u7ebf\u7684\u4e24\u4e2a\u6620\u5c04$\\Phi_0$\u548c$\\Phi_1$, \u8fd9\u91cc, \u540c\u4f26\u76f8\u5bf9\u4e8e\u8fb9\u754c$\\partial I=\\set{0,1}$(\u5b83\u7684\u50cf\u5728$\\Phi_0$\u548c$\\Phi_1$\u4e0b\u662f\u4e24\u6761\u9000\u5316\u7684\u66f2\u7ebfN\u548cS)\" src=\"http:\/\/i.imgur.com\/C9cpt.jpg\" alt=\"\u4ece $I^1=I$\u5230$S^2$\u4e0a\u66f2\u7ebf\u7684\u4e24\u4e2a\u6620\u5c04$\\Phi_0$\u548c$\\Phi_1$, \u8fd9\u91cc, \u540c\u4f26\u76f8\u5bf9\u4e8e\u8fb9\u754c$\\partial I=\\set{0,1}$(\u5b83\u7684\u50cf\u5728$\\Phi_0$\u548c$\\Phi_1$\u4e0b\u662f\u4e24\u6761\u9000\u5316\u7684\u66f2\u7ebfN\u548cS)\" width=\"280\" height=\"225\" \/><\/a><figcaption class=\"wp-caption-text\">\u4ece $I^1=I$\u5230$S^2$\u4e0a\u66f2\u7ebf\u7684\u4e24\u4e2a\u6620\u5c04${\\Phi_0}$\u548c${\\Phi_1}$, \u8fd9\u91cc, \u540c\u4f26\u76f8\u5bf9\u4e8e\u8fb9\u754c$\\partial I=\\{0,1\\}$(\u5b83\u7684\u50cf\u5728$\\Phi_0$\u548c$\\Phi_1$\u4e0b\u662f\u4e24\u6761\u9000\u5316\u7684\u66f2\u7ebfN\u548cS)<\/figcaption><\/figure>\n<p>$\\Phi_0$\u7684\u540c\u4f26\u7c7b$\\Pi$\u662f\u6240\u6709\u540c\u4f26\u4e8e$\\Phi_0$\u7684$\\Phi_1$\u6240\u6210\u4e4b\u96c6\u5408, \u800c$\\Pi$\u7684\u5bbd\u5ea6$L(\\Pi)$\u662f<br \/>\n\\[<br \/>\nL(\\Pi) = \\inf\\limits_{\\Phi_1\\in\\Pi}\\sup\\limits_{x\\in I^n} \\textrm{area}(\\Phi_1(x))<br \/>\n\\]<\/p>\n<p><span id='rem:2'><\/span><div class='latex_rem'><span class='latex_rem_h'>\u6ce8\u8bb0 2<\/span><span class='latex_rem_h'>.<\/span> \u5728Fernando \u548c Andr\u00e9\u6587\u7ae0\u4e2d, \u540c\u4f26\u7c7b(\u4ee5\u53ca\u4ed6\u4eec\u7684\u5bbd\u5ea6)\u7684\u6b63\u5f0f\u5b9a\u4e49\u4f9d\u8d56\u4e8e\u4e00\u4e9b\u51e0\u4f55\u6d4b\u5ea6\u8bba\u7684\u6982\u5ff5, \u4f8b\u5982(\u79ef\u5206)\u6d41(integral currents), \u53d8\u5c42(varifolds), \u4ee5\u53ca\u79ef\u5206\u6d41\u4e2d\u7684\u5e73\u5766\u5ea6\u91cf\u548c\u5e73\u5766\u62d3\u6251(\u57fa\u4e8e(mass)\u7684\u6982\u5ff5). \u7279\u522b\u5730, \u4e0a\u9762\u63d0\u5230\u7684\u6620\u5c04$\\Phi$\u5c06\u6bcf\u4e00\u4e2a$x\\in I^n$\u4e0e\u4e00\u4e2a\u6ca1\u6709\u8fb9\u754c\u7684\u79ef\u5206\u6d41$\\Phi(x)$\u5bf9\u5e94, \u800c\u4e14\u5047\u8bbe$\\Phi$\u5728\u5e73\u5766\u62d3\u6251\u4e0b\u662f\u8fde\u7eed\u7684. \u7c7b\u4f3c\u5730, \u540c\u4f26$\\Psi$\u5728\u8d28\u91cf\u62d3\u6251(mass topology)\u4e0b\u4e5f\u662f\u8fde\u7eed\u7684. \u8be6\u60c5\u8bf7\u53c2\u8003 Fernando \u548c Andr\u00e9[<a href='#MarquesNeves2014Min'>5<\/a>]\u7684\u7b2c4, 7\u8282.<br \/>\n<\/div><br \/>\n<div class='latex_examp'><span class='latex_examp_h'>\u4f8b\u5b50 1<\/span><span class='latex_examp_h'>.<\/span> \u5bf9${s\\in I=[0,1]}$, ${\\Phi_{(0)}(s):=\\{x_4=2s-1\\}\\subset S^3}$\u7684$\\Pi_0$\u540c\u4f26\u7c7b(\u7c97\u7565\u5730\u53ef\u770b\u6210\u4e0a\u9762\u7b2c\u4e8c\u4e2a\u56fe\u4e2d\u84dd\u5706\u5708\u6240\u5bf9\u5e94\u7684\u66f2\u7ebf\u7c07), \u53ef\u4ee5\u8bc1\u660e(\u76f4\u89c2\u5730)${L(\\Pi_0)=4\\pi}$.<br \/>\n<\/div><br \/>\n<div class='latex_rem'><span class='latex_rem_h'>\u6ce8\u8bb0 3<\/span><span class='latex_rem_h'>.<\/span> \u4ece\u67d0\u79cd\u610f\u4e49\u4e0a\u8bf4, \u4e0a\u4f8b\u4e2d\u7684\u540c\u4f26\u7c7b$\\Pi_0$\u53ef\u4ee5\u770b\u6210\u540c\u4f26\u7fa4${\\pi_1(\\mathcal{S},0)}$\u4e2d\u7684\u4e00\u4e2a\u5143\u7d20, \u5176\u4e2d$\\mathcal{S}$\u662f$S^3$\u4e2d\u5168\u4f53\u66f2\u9762\u7ec4\u6210\u7684\u7a7a\u95f4, \u800c${0}$ \u662f\u5e73\u51e1\u66f2\u9762(\u9762\u79ef\u4e3a\u96f6).<br \/>\n<\/div><br \/>\n\u5728$\\Sigma$\u662f\u5d4c\u5165\u5230$S^3$\u4e2d\u4e8f\u683c\u4e3a$g\\geq1$\u7684\u95ed\u66f2\u9762\u7684\u60c5\u5f62, Fernando\u548cAndr\u00e9\u5173\u8054\u4e86\u67d0\u4e2a5-\u7ef4\u6781\u5c0f-\u6781\u5927\u540c\u4f26\u7c7b$\\Pi$(\u5373, \u4ed6\u4eec\u8003\u8651\u4e86\u5173\u4e8e$\\Sigma$\u7684\u67d0\u4e2a\u5b9a\u4e49\u5728$I^5$\u4e0a\u7684\u540c\u4f26\u7c7b$\\Pi$)\u4ed6\u4eec\u8bc1\u660e\u5982\u4e0b\u7684\u6781\u5c0f-\u6781\u5927\u5b9a\u7406(min-max Theorem):<br \/>\n<span id='thm:min-max'><\/span><div class='latex_thm'><span class='latex_thm_h'>\u5b9a\u7406 3<\/span> (<span class='latex_thm_name'>F. C. Marques, A. Neves<\/span>)<span class='latex_thm_h'>.<\/span> \u4ee4$\\Sigma\\subset S^3$\u662f\u4e00\u4e2a\u4e8f\u683c\u4e3a$g\\geq1$\u7684\u5d4c\u5165\u95ed\u66f2\u9762. \u5219\u5b58\u5728\u4e00\u4e2a\u6781\u5c0f\u5d4c\u5165\u66f2\u9762${\\widehat{\\Sigma}\\subset S^3}$\u5176\u4e8f\u683c$g\\geq1$(\u53ef\u80fd\u4e0d\u662f\u8fde\u901a\u7684, \u800c\u4e14\u53ef\u80fd\u6709\u91cd\u6570(multiplicities))\u4f7f\u5f97<br \/>\n\\[<br \/>\n4\\pi&lt; \\textrm{area}(\\widehat{\\Sigma})=L(\\Pi)\\leq\\mathcal{W}(\\Sigma)<br \/>\n\\]<br \/>\n\u5176\u4e2d$\\Pi$\u662f$\\Sigma$\u7684\u6781\u5c0f-\u6781\u5927\u540c\u4f26\u7c7b(min-max homotopy class of ${\\Sigma}$).<br \/>\n<\/div><br \/>\n\u6b63\u5982\u6211\u4eec\u5df2\u7ecf\u63d0\u5230\u7684, \u8fd9\u5c31\u4f7f\u5f97\u4ed6\u4eec\u53ef\u4ee5\u5c06<a class='latex_ref' href=#><span style=\"colro:red\">thm2<\/span><\/a>\u5173\u4e8e\u4e00\u822c${\\Sigma\\subset S^3}$\u7684\u8bc1\u660e\u5212\u5f52\u4e3a\u5bf9\u6781\u5c0f\u66f2\u9762${\\widehat{\\Sigma}\\subset S^3}$\u9762\u79ef\u7684\u7814\u7a76. \u8fd9\u91cc, Fernando\u548cAndr\u00e9\u8bc1\u660e\u4e86<br \/>\n<span id='thm:cor'><\/span><div class='latex_thm'><span class='latex_thm_h'>\u5b9a\u7406 4<\/span> (<span class='latex_thm_name'>F. C. Marques,A. Neves<\/span>)<span class='latex_thm_h'>.<\/span> \u4ee4${\\widehat{\\Sigma}\\subset S^3}$\u662f\u4e00\u4e2a\u4e8f\u683c$g\\geq1$\u7684\u5d4c\u5165\u3001\u95ed\u3001\u6781\u5c0f\u66f2\u9762. \u5219<br \/>\n\\[<br \/>\n\\textrm{area}(\\widehat{\\Sigma})\\geq 2\\pi^2.<br \/>\n\\]<br \/>\n\u800c\u4e14${\\textrm{area}(\\widehat{\\Sigma})=2\\pi^2}$\u6210\u7acb, \u5f53\u4e14\u4ec5\u5f53${\\widehat{\\Sigma}}$\u548cClifford\u73af\u9762${\\Sigma_c}$\u7b49\u8ddd.<br \/>\n<\/div><br \/>\n\u5f53\u7136, \u5229\u7528<a class='latex_ref' href=#thm:min-max>\u5b9a\u7406 3<\/a>\u548c<a class='latex_ref' href=#thm:cor>\u5b9a\u7406 4<\/a>, <a class='latex_ref' href=#thm:main>\u5b9a\u7406 2<\/a>\u4e2d\u7684\u4e0d\u7b49\u5f0f${\\mathcal{W}(\\Sigma)\\geq 2\\pi^2}$&#8221;\u51e0\u4e4e&#8221;\u6210\u7acb. \u4e8b\u5b9e\u4e0a, \u6211\u4eec\u8bf4&#8221;\u51e0\u4e4e&#8221;, \u662f\u56e0\u4e3a\u5982\u679c\u4e8f\u683c$g\\geq1$\u7684\u6781\u5c0f\u66f2\u9762${\\widehat{\\Sigma}}$\u662f\u8fde\u901a\u7684, \u800c\u4e14\u91cd\u6570\u4e3a1, \u90a3\u4e48\u5c31\u53ef\u4ee5\u5f97\u5230<a class='latex_ref' href=#thm:willmore_conj>1<\/a>. \u5f53\u7136, \u8fd9\u5728\u4e00\u822c\u60c5\u5f62\u4e0d\u4e00\u5b9a\u6b63\u786e, \u4f46\u662f, \u6211\u4eec\u5c06\u5728\u8fd9\u7bc7\u65e5\u5fd7\u7684\u7ed3\u5c3e\u5904\u770b\u5230, <a class='latex_ref' href=#thm:main>2<\/a>\u53ef\u4ee5\u5f88\u5bb9\u6613\u5730\u4ece<a class='latex_ref' href=#thm:min-max>3<\/a>\u548c<a class='latex_ref' href=#thm:cor>4<\/a>\u63a8\u51fa. \u4f46\u540c\u65f6, ${\\mathcal{W}(\\Sigma)=2\\pi^2}$\u53ef\u4ee5\u8868\u5f81Clifford\u73af\u9762$\\Sigma_c$\u5e76\u4e0d\u662f\u50cf\u4e0a\u9762\u53d9\u8ff0\u7684\u90a3\u6837\u53ef\u4ee5\u4f5c\u4e3a<a class='latex_ref' href=#thm:min-max>3<\/a>\u548c<a class='latex_ref' href=#thm:cor>4<\/a>\u7684\u76f4\u63a5\u63a8\u8bba, \u5c3d\u7ba1\u5982\u6b64, \u6211\u4eec\u5c06\u770b\u5230\u5b83\u5176\u5b9e\u662f\u8574\u542b\u5728\u8fd9\u4e9b\u5b9a\u7406\u7684\u8bc1\u660e\u4e4b\u4e2d.<\/p>\n<p>\u5728\u4efb\u4f55\u60c5\u51b5\u4e0b, \u5728\u8bb2\u5b8c\u8fd9\u4e9b\u6709Fernando \u548c Andr\u00e9\u5f97\u5230\u7684\u4e3b\u8981\u7ed3\u679c\u7684\u4eae\u70b9\u4e4b\u540e, \u6211\u4eec\u8f6c\u5165\u5bf9<a class='latex_ref' href=#thm:main>2<\/a>, <a class='latex_ref' href=#thm:min-max>3<\/a>\u548c<a class='latex_ref' href=#thm:cor>4<\/a>\u7684\u8bc1\u660e\u7684\u4e00\u822c\u7b56\u7565\u7684\u63cf\u8ff0(\u8bf7\u540c\u65f6\u53c2\u8003\u4ed6\u4eec\u6587\u7ae0\u7684\u7b2c2\u8282).<br \/>\n<span class=\"latex_section\">2.&#x00A0;\u5b9a\u74063\u7684\u8bc1\u660e\u7b56\u7565<a id=\"sec:2\"><\/a><\/span>\n\nFernando\u4e0eAndr\u00e9\u6587\u7ae0\u7684\u5f88\u5927\u4e00\u90e8\u5206\u90fd\u7528\u6765\u6784\u9020Almgren-Pitts\u7684\u6781\u5c0f&#8212;\u6781\u5927\u7406\u8bba, \u5176\u76ee\u7684\u662f\u4f7f\u5f97\u67d0\u4e2a\u540c\u4f26\u7c7b\u7684\u5bbd\u5ea6\u53ef\u4ee5\u7531\u6781\u5c0f\u66f2\u9762\u7684\u9762\u79ef\u8868\u793a. \u5728\u63a5\u4e0b\u6765\u7684\u4e00\u5c0f\u8282, \u6211\u4eec\u53d9\u8ff0\u4e86\u8fd9\u4e2a\u6781\u5c0f\u6781\u5927\u7406\u8bba\u7684\u4e00\u4e2a\u5178\u578b\u60c5\u51b5.<br \/>\n<span class=\"latex_subsection\">2.1.&#x00A0;\u6781\u5c0f&#8212;\u6781\u5927\u7406\u8bba\u7684\u5178\u578b\u4f8b\u5b50<a id=\"sec:2.1\"><\/a><\/span>\n\n\u4ee4 ${M=M^3}$ \u662f\u7d27\u81f43\u7ef4\u6d41\u5f62$\\Pi$\u662f\u8fde\u7eed\u6620\u5c04${\\Phi}$\u7684\u540c\u4f26\u7c7b, \u5176\u4e2d$\\Phi$\u5c06\u6bcf\u4e2a${x\\in I^n=[0,1]^n}$\u5bf9\u5e94\u4e8e$M$\u4e2d\u4e00\u7d27\u81f4\u66f2\u9762.<br \/>\n<span id='thm:min-max-approx'><\/span><div class='latex_thm'><span class='latex_thm_h'>\u5b9a\u7406 5<\/span> (<span class='latex_thm_name'>Min-Max\u5b9a\u7406<\/span>)<span class='latex_thm_h'>.<\/span> \u5047\u8bbe<br \/>\n\\[<br \/>\nL(\\Pi)>\\sup\\limits_{x\\in\\partial I^n}\\textrm{area}(\\Phi(x)).<br \/>\n\\]<br \/>\n\u5219\u5b58\u5728\u5149\u6ed1\u5d4c\u5165\u6781\u5c0f\u66f2\u9762${\\Sigma\\subset M}$(\u53ef\u80fd\u4e0d\u8fde\u901a\u4e14\u6709\u91cd\u6570)\u4f7f\u5f97<br \/>\n\\[<br \/>\n\\textrm{area}(\\Sigma)=L(\\Pi).<br \/>\n\\]<br \/>\n\u6b64\u5916, \u7ed9\u5b9a${\\{\\Phi_i\\}_{i\\in\\mathbb{N}}\\subset\\Pi}$, \u6ee1\u8db3<br \/>\n\\[<br \/>\n\\lim\\limits_{i\\rightarrow\\infty} \\sup\\limits_{x\\in I^n} \\textrm{area}(\\Phi_i(x)) = L(\\Pi),<br \/>\n\\]<br \/>\n\u6211\u4eec\u53ef\u4ee5\u9009\u62e9${x_i\\in I^n}$, ${i\\in\\mathbb{N}}$, \u4f7f\u5f97<br \/>\n\\[<br \/>\n\\Sigma=\\lim\\limits_{i\\rightarrow\\infty}\\Phi_i(x_i).<br \/>\n\\]<br \/>\n<\/div><br \/>\n\u521d\u7565\u6765\u8bb2, \u4e0a\u4e00\u5b9a\u7406\u4e2d\u7684\u6761\u4ef6${L(\\Pi)>\\sup\\limits_{x\\in\\partial I^n}\\textrm{area}(\\Phi(x))}$\u53ef\u770b\u4f5c\u5728\u8fb9\u754c\u7684\u67d0\u79cd\u969c\u788d. \u7528\u8fd9\u79cd\u8bed\u8a00, \u4e0a\u9762\u7684\u6781\u5c0f&#8212;\u6781\u5927\u5b9a\u7406\u662f\u8bf4\u5229\u7528\u6781\u5c0f\u66f2\u9762\u7684\u9762\u79ef\u6765\u5b9e\u73b0\u540c\u4f26\u7c7b$\\Pi$\u7684\u5bbd\u5ea6$L(\\Pi)$\u8fd9\u4e00\u53d8\u5206\u95ee\u9898\u53ef\u4ee5\u901a\u8fc7\u53d6\u4e00\u5217\u6070\u5f53\u7684\u66f2\u9762$\\Phi_i(x_i)$\u7684\u6781\u9650\u6765\u89e3\u51b3(\u4f7f\u5f97\u66f2\u9762\u7684\u9762\u79ef\u8d8b\u4e8e$L(\\Pi)$),  \u7531\u4e8e\u969c\u788d\u6761\u4ef6$\\Phi_i(x_i)$\u6392\u9664\u4e86$\\Phi_i(x_i)$&#8221;\u9003\u51fa&#8221;\u8fb9\u754c\u7684\u53ef\u80fd, \u56e0\u800c${\\Phi_i(x_i)}$\u5fc5\u5b9a\u6536\u655b.<br \/>\n\u5f53\u7136, \u4e0a\u9762\u5b9a\u7406\u7684\u9648\u8ff0\u53ea\u662fFernando &amp; Andr\u00e9\u8bc1\u660e\u7684\u6781\u5c0f\u6781\u5927\u5b9a\u7406\u7684\u7c97\u7565\u7248\u672c. \u66f4\u7cbe\u786e\u7684\u9648\u8ff0\u53ef\u4ee5\u53c2\u8003\u4ed6\u4eec\u6587\u7ae0\u7684\u5b9a\u74068.5(\u4ee5\u53ca\u547d\u98988.4). \u6211\u4eec\u5df2\u7ecf\u63d0\u53ca, \u6781\u5c0f\u6781\u5927\u5b9a\u7406\u7684\u8bc1\u660e(\u6216\u8005, \u66f4\u6070\u5f53\u7684, \u547d\u98988.4\u7684\u8bc1\u660e)\u975e\u5e38\u957f\u800c\u4e14\u6280\u5de7\u5f3a, \u5360\u4e86\u4ed6\u4eec\u6587\u7ae0\u5f88\u5927\u7684\u4e00\u90e8\u5206(\u537313, 14, 15\u8282\u4ee5\u53ca\u9644\u5f55C). \u7279\u522b\u5730, \u5173\u4e8e\u8fd9\u90e8\u5206\u7ec6\u8282\u7684\u8ba8\u8bba\u5df2\u8d85\u51fa\u672c\u6587\u7684\u76ee\u7684, \u6211\u4eec\u5728\u4e0b\u9762\u7684\u8ba8\u8bba\u4e2d\u5c06\u4e0d\u52a0\u8bc1\u660e\u5730\u627f\u8ba4\u8be5\u5b9a\u7406.<br \/>\n<div class='latex_rem'><span class='latex_rem_h'>\u6ce8\u8bb0 4<\/span><span class='latex_rem_h'>.<\/span> \u5728Fernando &amp; Andr\u00e9 \u7684\u6587\u7ae0\u4e2d\u63d0\u5230, \u901a\u8fc7\u7c7b\u4f3c\u4e8eMorse\u7406\u8bba, \u4e0a\u8ff0\u6781\u5c0f\u6781\u5927\u5b9a\u7406\u4e2d\u7684\u6781\u5c0f\u66f2\u9762$\\Sigma$\u7684\u6307\u6807\u5e94\u8be5$\\leq n$(\u56e0\u4e3a\u5b83\u6765\u81f3\u4e8e\u4e00\u7c07$n$-\u53c2\u6570\u66f2\u9762). \u4e00\u822c\u5730, \u8fd9\u4e00\u7cbe\u7ec6\u7684\u8bba\u65ad\u4e0d\u597d\u9a8c\u8bc1, \u4f46\u662f, \u5c3d\u7ba1\u5982\u6b64, \u6211\u4eec\u5c06(\u5728<a class='latex_ref' href=#thm:cor>\u5b9a\u7406 4<\/a>\u7684\u8bc1\u660e\u4e2d)\u770b\u5230\u5728Clifford\u73af\u9762($n=5$)\u7684\u60c5\u5f62\u8fd9\u4e00\u65ad\u8a00\u786e\u5b9e\u6b63\u786e.<br \/>\n<\/div><br \/>\n\u4e00\u65e6\u6211\u4eec\u5047\u5b9a\u6781\u5c0f\u6781\u5927\u5b9a\u7406, \u6211\u4eec\u5c06\u7ee7\u7eed\u5728\u63a5\u4e0b\u6765\u7684\u4e09\u5c0f\u8282\u4e2d\u7ed9\u51fa<a class='latex_ref' href=#thm:min-max>\u5b9a\u7406 3<\/a>\u7684\u6982\u8981: \u5373\u6784\u9020\u5d4c\u5165\u5230$S^3$\u7684\u7d27\u81f4\u66f2\u9762$\\Sigma$\u7684\u6781\u5c0f\u6781\u5927\u540c\u4f26\u7c7b.<br \/>\n<span class=\"latex_subsection\">2.2.&#x00A0;\u5178\u5219\u66f2\u9762\u7c07<a id=\"sec:2.2\"><\/a><\/span>\n\n\u4ee4$B^4=\\{x\\in\\mathbb{R}^4: |x|<1\\}$\u662f\u4e00\u4e2a\u5f00\u5355\u4f4d\u7403. \u5bf9\u4efb\u4f55${v\\in B^4}$, \u4ee4 ${F_v:S^3\\rightarrow S^3}$\u662f\u5982\u4e0b\u5b9a\u4e49\u7684\u5171\u5f62\u6620\u5c04\n\\[\nF_v(x):=\\frac{(1-|v|^2)}{|x-v|^2}(x-v) -x\n\\]\n\u7ed9\u5b9a\u5149\u6ed1\u7d27\u66f2\u9762${\\Sigma\\subset S^3}$, \u6211\u4eec\u6709\u5982\u4e0b\u5b9a\u4e49\u7684\u5178\u5219\u66f2\u9762\u7c07\n\\[\n\\Sigma_{(v,t)}:=\\partial\\{x\\in S^3: d_v(x)<t\\}, \\quad (v,t)\\in B^4\\times [-\\pi,\\pi].\n\\]\n\u8fd9\u91cc, ${d_v(x)}$\u662f$x$\u5230\u66f2\u9762${F_v(\\Sigma)}$\u7684\u6709\u5411\u8ddd\u79bb(\u5728$S^3$\u7684\u6807\u51c6\u7403\u5ea6\u91cf$d$\u4e0b), \u5373\u901a\u8fc7\u9009\u53d6$\\Sigma$\u7684\u4e00\u4e2a\u5b9a\u5411, \u6211\u4eec\u53ef\u4ee5\u5c06$S^3-\\Sigma$\u89c6\u4e3a$A^{int}\\cup A^{ext}$\u5176\u4e2d$A^{int}$\u662f$\\Sigma$\u7684\u5185\u90e8($S^3-\\Sigma$\u7684\u8fde\u901a\u90e8\u5206), \u800c${A^{ext}}$\u662f$\\Sigma$\u7684\u5916\u90e8(${S^3-\\Sigma}$\u53e6\u4e00\u8fde\u901a\u5206\u652f), \u6211\u4eec\u4ee4\n\\[\nd_v(x) = \\left\\{\\begin{array}{cl}d(x,F_v(\\Sigma)) &#038; \\textrm{if } x\\in F_v(A^{ext}) \\\\ -d(x,F_v(\\Sigma)) &#038; \\textrm{if } x\\in F_v(A^{int})\\end{array}\\right.\n\\]\n\\begin{rem}\n\u56e0\u4e3a$S^3$\u7684\u76f4\u5f84\u4e3a$\\pi$, \u6211\u4eec\u6709${\\Sigma_{v,-\\pi}}$\u4ee5\u53ca${\\Sigma_{v,\\pi}}$\u662f\u5e73\u51e1\u66f2\u9762(\u9762\u79ef\u4e3a\u96f6), \u8fd9\u91cc${v\\in B^4}$\u662f\u4efb\u610f\u7684. \n\n\u7531\u4e8e${F_v}$\u662f\u5171\u5f62\u7684, \u6211\u4eec\u6709${\\mathcal{W}(F_v(\\Sigma))=\\mathcal{W}(\\Sigma)}$, \u5bf9\u4efb\u610f\u7684${v\\in B^4}. \u53e6\u4e00\u65b9\u9762A. Ros[<a href='#Ros1999Willmore'>8<\/a>]\u7684\u4e00\u4e2a\u7ed3\u679c\u8868\u660e\n\\[\n\\textrm{area}(\\Sigma_{(v,t)})\\leq\\mathcal{W}(\\Sigma) - \\frac{\\sin^2t}{2}\\int_{\\Sigma} |\\mathring{A}|^2 d\\Sigma\\leq\\mathcal{W}(\\Sigma)\n\\]\n\u5176\u4e2d${\\mathring{A}}$\u662f$\\Sigma$\u7684\u7b2c\u4e8c\u57fa\u672c\u578b$A$\u7684\u65e0\u8ff9\u90e8\u5206. \u4e8b\u5b9e\u4e0a, \u4e0a\u9762\u8fd9\u4e2a\u4f30\u8ba1\u7684\u8bc1\u660e\u4e0d\u96be: \u5b83\u53ef\u901a\u8fc7\u76f4\u63a5\u8ba1\u7b97$F_v$(\u53ea\u6709\u4e00\u9875\u7eb8, \u53ef\u4ee5\u53c2\u8003Fernando &amp; Andr\u00e9\u6587\u7ae015\u9875\u5b9a\u74063.4).\n\n\u7efc\u4e0a, \u7ed9\u5b9a\u7d27\u5d4c\u5165\u66f2\u9762$\\Sigma\\subset S^3$, \u6211\u4eec\u6709\u5178\u52195\u7ef4\u66f2\u9762\u7c07$\\Sigma_{(v,t)}$, $(v,t)\\in B^4\\times[-\\pi,\\pi]$\u4f7f\u5f97\n\\begin{equation}\\label{eq:1}\n\\textrm{area}(\\Sigma_{(v,t)})\\leq \\mathcal{W}(\\Sigma)=\\mathcal{W}(F_v(\\Sigma)) \n\\end{equation}\n\u9274\u4e8e\u8fd9\u4e2a\u4f30\u8ba1\u4ee5\u53ca<a class='latex_ref' href=#thm:min-max>\u5b9a\u7406 3<\/a>\u7684\u9648\u8ff0, \u5c06<a class='latex_ref' href=#thm:min-max-approx>\u5b9a\u7406 5<\/a>\u5e94\u7528\u4e8e\u8fd9\u7c07\u5178\u5219\u66f2\u9762$\\Sigma_{(v,t)}$, $(v,t)\\in B^4\\times[-\\pi,\\pi]$\u662f\u975e\u5e38\u6709\u7406\u7531\u7684. \u7136\u800c\u8fd9\u5e76\u4e0d\u884c: $B^4$ \u4e0d\u662f\u7d27\u81f4\u7684, \u800c\u4e14\u4efb\u4f55\u5408\u7406\u7684\u5c06\u5178\u5219\u7c07\u5ef6\u62d3\u5230$\\overline{B}^4\\times [-\\pi,\\pi]\\simeq I^5$\u4e0a\u90fd\u4e0d\u662f\u8fde\u7eed\u7684, \u5373, \u5728\u5e73\u5766\u62d3\u6251\u4e0b(\u53c2\u8003<a class='latex_ref' href=#rem:2>\u6ce8\u8bb0 2<\/a>), \u4f7f\u5f97\u5982\u90a3\u6837\u5ef6\u62d3\u540e\u7684\u5178\u5219\u7c07\u4e0d\u80fd\u5b9a\u4e49\u5408\u7406\u7684\u540c\u4f26\u7c7b. \u4f8b\u5982, $F_v(\\Sigma)$ \u4e0e $\\Sigma_{(v,t)}$ \u5bf9 $B^4\\ni v\\rightarrow p\\in S^3-\\Sigma=\\partial B^4-\\Sigma$, \u4ecd\u7136\u6709\u5b9a\u4e49, \u6bd4\u5982 ${\\Sigma_{(v,t)}\\rightarrow\\partial B_{\\pi+t}(p)}$ \u6216\u8005 $\\partial B_t(-p)$(\u8fd9\u91cc\u50cf\u96c6\u7684\u53d6\u6cd5\u53d6\u51b3\u4e8e${v\\rightarrow p\\in S^3-\\Sigma=A^{int}\\cup A^{ext}}$\u662f\u4f9d\u8d56\u4e8e${p\\in A^{ext}}$ \u8fd8\u662f${A^{int}}$, \u540c\u6837\u7684\u5ef6\u62d3\u5bf9$B^4\\ni v\\rightarrow p\\in\\Sigma$ \u4e0d\u518d\u6210\u7acb, \u8fd9\u662f\u56e0\u4e3a${\\Sigma_{(v,t)}}$\u5728$v\\rightarrow p\\in\\Sigma$\u65f6\u7684\u6781\u9650\u4f9d\u8d56\u4e8e\"\u6536\u655b\u7684\u89d2\u5ea6\": \u66f4\u51c6\u786e\u7684\u8bf4, \u4ee4$N(p)$\u4e3a$\\Sigma\\subset S^3$\u5728$p$\u5904\u7684\u5355\u4f4d\u5411\u91cf, \u5982\u679c\n\\[\nv_n=|v_n|(\\cos(s_n)p+\\sin(s_n)N(p))\n\\]\n\u8d8b\u4e8e$p\\in \\Sigma$ (\u5373$|v_n|<1$, $|v_n|\\rightarrow 1$ \u4e14 $s_n\\rightarrow 0$, $n\\rightarrow\\infty$), \u5219\n\\[\n\\Sigma_{(v_n,t)}\\rightarrow \\partial B_{\\frac{\\pi}{2}-\\theta+t}(-\\sin(\\theta)p-\\cos(\\theta)N(p))\n\\]\n\u5176\u4e2d ${\\theta=\\lim\\limits_{n\\rightarrow\\infty}\\arctan s_n\/(1-|v_n|)\\in [-\\pi\/2,\\pi\/2]}$. \u8bf7\u770b\u5982\u4e0b\u7684\u793a\u610f\u56fe(\u4f4e\u7ef4): \u4e00\u5217$v_n$\u8d8b\u4e8e\u4f4d\u4e8e$\\Sigma$\u5168\u6d4b\u5730\u7403(\u8d64\u9053)\u7684$p$.\n<a href=\"https:\/\/matheuscmss.files.wordpress.com\/2012\/04\/geom4.jpg?w=560&#038;h=488\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/matheuscmss.files.wordpress.com\/2012\/04\/geom4.jpg?w=560&#038;h=488\" width=\"560\" height=\"488\" class=\"aligncenter size-medium\" \/><\/a><br \/>\n\u5173\u4e8e\u5f53${v\\rightarrow S^3=\\partial B^4}$\u65f6${\\Sigma_{(v,t)}}$\u884c\u4e3a\u7684\u8be6\u7ec6\u8ba8\u8bba\u5728Fernando &amp; Andr\u00e9\u6587\u7ae0\u7b2c5\u8282(\u53c2\u8003\u547d\u98985.3, \u5f15\u74065.4 \u4ee5\u53ca\u9644\u5f55B).<\/p>\n<p>\u65e0\u8bba\u5728\u90a3\u79cd\u60c5\u5f62, \u6211\u4eec\u5df2\u7ecf\u770b\u5230\u5c06\u5178\u5219\u7c07${\\Sigma_{(v,t)}}$\u5408\u7406\u5ef6\u62d3\u5230${\\overline{B^4}\\times [-\\pi,\\pi]}$\u7684\u4e3b\u8981\u969c\u788d\u662f\u96c6\u5408$\\Sigma$. \u8fd9\u91cc, Fernando &amp; Andr\u00e9\u901a\u8fc7&#8221;\u7206\u7834&#8221;(blowing up)$\\Sigma$\u6765\u89e3\u51b3, \u8fd9\u6b63\u662f\u6211\u4eec\u4e0b\u4e00\u8282\u7684\u8bdd\u9898.<br \/>\n<span class=\"latex_subsection\">2.3.&#x00A0;\u7206\u7834\u5904\u7406<a id=\"sec:2.3\"><\/a><\/span>\n\n\u56fa\u5b9a$\\varepsilon>0$\u5c0f\u4e14\u4ee4${\\Omega_{\\varepsilon}}$\u662f$\\Sigma$\u5728${\\overline{B^4}}$\u4e2d\u4e00\u4e2a\u534a\u5f84\u4e3a$\\varepsilon>0$\u7684\u7ba1\u72b6\u90bb\u57df, \u5373,<br \/>\n\\begin{multline*}<br \/>\n\\Omega_{\\varepsilon}=\\Bigg\\{\\Lambda(p,s_1,s_2) :=(1-s_1)(\\cos(s_2)p+\\sin(s_2)N(p));\\\\<br \/>\n|(s_1,s_2)| :=\\sqrt{s_1^2+s_2^2}<\\varepsilon, s_1\\geq 0\\Bigg\\}\n\\end{multline*}\n\u7136\u540e, \u8003\u8651\u8fde\u7eed\u6620\u5c04${T:\\overline{B^4}\\rightarrow\\overline{B^4}}$\u4f7f\u5f97:\n<ul>\n<li>$T$ \u662f${B^4-\\overline{\\Omega_{\\varepsilon}}}$ \u5230 $B^4$\u7684\u540c\u80da, \u4e14\n<\/li><li>$T$ \u5c06 ${\\overline{\\Omega_{\\varepsilon}}}$ \u6620\u5c04\u4e3a $\\Sigma$, \u8fd9\u91cc\u6620\u5c04\u662f\u6700\u8fd1\u70b9\u6295\u5c04(projection to the nearest point).\n<\/li><\/ul>\n\u66f4\u52a0\u6b63\u786e\u5730, \u6211\u4eec\u53ef\u4ee5\u5b9a\u4e49$T$\u4e3a${\\overline{B^4}-\\Omega_{3\\varepsilon}}$\u4e0a\u7684\u6052\u7b49\u6620\u5c04\u4e14$T(\\Lambda(p,s_1,s_2)) = \\Lambda(p,\\phi(|(s_1,s_2)|)s_1,\\phi(|(s_1,s_2)|)s_2)$ \u5176\u4e2d $\\phi$ \u662f\u4e00\u5149\u6ed1\u622a\u65ad\u51fd\u6570(bump function), $\\phi$ \u5728$[0,\\varepsilon]$\u7b49\u4e8e0, $\\phi$ \u5728 $[\\varepsilon,2\\varepsilon]$\u4e0a\u4e25\u683c\u9012\u589e, \u4e14 $\\phi$ \u5728 $[2\\varepsilon,3\\varepsilon]$\u4e0a\u7b49\u4e8e1.\n\n\n\u5728\u4e0b\u9762, \u6211\u4eec\u7ed9\u51fa\u4e86\u5f53$\\Sigma$\u662f\u5b50\u5348\u7ebf(\u5168\u6d4b\u5730\u7403)\u65f6${B^4-\\overline{\\Omega_{\\varepsilon}}}$\u7684\u4e00\u4e2a\u56fe\u793a, \u8bfb\u8005\u5c06\u4f1a(\u975e\u5e38\u5bb9\u6613\u5730)\u76f4\u89c2\u7684\u4e86\u89e3\u7206\u7834\u6620\u5c04$T$\u7684\u4f5c\u7528.\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/matheuscmss.files.wordpress.com\/2012\/04\/geom3.jpg\" width=\"482\" height=\"482\" class=\"alignnone size-medium\" \/><\/p>\n<p>\u5229\u7528$T$, Fernando &amp; Andr\u00e9 \u4fee\u6539\u4e86\u6700\u521d\u7684\u5178\u5219\u66f2\u9762\u7c07$\\Sigma_{(v,t)}$, \u5176\u65b9\u6cd5\u662f\u5bf9\u6bcf\u4e2a$(v,t)\\in (B^4-\\overline{\\Omega_{\\varepsilon}})\\times [-\\pi,\\pi]$\u5f15\u5165\u66f2\u9762\u7c07$C(v,t)=\\Sigma_{(T(v),t)}$. \u8fd9\u91cc, \u7531\u4e8e\u6211\u4eec\u5272\u6389\u4e86$\\Sigma$\u7684\u4e00\u4e2a$\\Omega_{\\varepsilon}$\u7ba1\u72b6\u90bb\u57df, \u4ece$T$\u7684\u7279\u6027\u77e5\u9053, \u5229\u7528$\\Sigma_{(v,t)}$\u800c\u4ea7\u751f\u7684&#8221;\u6536\u655b\u7684\u89d2\u5ea6&#8221;\u95ee\u9898\u53ef\u4ee5\u901a\u8fc7\u4f7f\u7528$C(v,t)$\u6765\u89e3\u51b3: $\\Omega_{\\varepsilon}$ \u4e0e$T$\u7684\u5b9a\u4e49(\u4e4b\u9009\u62e9)\u4fdd\u8bc1\u4e86\u5728\u9760\u8fd1$\\Sigma$\u4e4b\u524d\u5df2\u505c\u6b62, \u4ece\u800c\u9009\u62e9\u4e86\u4e00\u4e2a&#8221;\u56fa\u5b9a\u89d2\u5ea6\u7684\u6536\u655b&#8221;.<\/p>\n<p>\u7279\u522b\u5730, \u4e0d\u96be\u8bf4\u670d\u81ea\u5df1, \u65b0\u7684\u66f2\u9762\u7c07$C$\u5141\u8bb8\u4e00\u4e2a&#8221;\u5408\u7406\u7684&#8221;(\u5373<a class='latex_ref' href=#rem:2>\u6ce8\u8bb0 2<\/a>\u4e2d\u8bf4\u7684\u5728\u65e0\u8fb9\u7684\u79ef\u5206\u6d41\u7a7a\u95f4\u7684\u5e73\u5766\u62d3\u6251\u4e0b\u8fde\u7eed)\u5230$\\overline{B^4-\\Omega_{\\varepsilon}}\\times [-\\pi,\\pi]$\u7684\u5ef6\u62d3. \u7136\u540e, \u6211\u4eec\u53ef\u4ee5\u8fdb\u4e00\u6b65\u5c06$C$\u5ef6\u62d3\u5230$\\Omega_{\\varepsilon}$\u4f7f\u5f97$C$\u5728\u5f84\u5411\u65b9\u5411\u662f\u5e38\u503c, \u5373$C(\\Lambda(p,s_1,s_2)):=C(\\Lambda(p,\\varepsilon,s_2))$.<\/p>\n<p>\u901a\u8fc7\u8fd9\u79cd\u65b9\u5f0f, Fernando &amp; Andr\u00e9\u8bc1\u660e(\u53c2\u8003\u4ed6\u4eec\u6587\u7ae0\u4e2d\u5b9a\u74065.1)\u53ef\u5f97\u5230\u5b9a\u4e49\u5728$\\overline{B^4}\\times[-\\pi,\\pi]$\u4e0a\u7684\u4e00\u7c07&#8221;\u5408\u7406\u7684&#8221;\u66f2\u9762$C$, \u4f7f\u5f97<br \/>\n<ul><li>\u5bf9\u6240\u6709$v\\in\\overline{B^4}$,$\\textrm{area}(C(v,-\\pi))=\\textrm{area}(C(v,\\pi))=0$;<\/li><li>\u5f53$v\\in S^3\\cup\\overline{\\Omega_{\\varepsilon}}$\u65f6, $C(v,t)$ \u662f $S^3$ \u4e2d\u7684\u6d4b\u5730\u7403;<\/li><li>\u5bf9\u6bcf\u4e2a$v\\in S^3$, \u5b58\u5728\u552f\u4e00\u7684$s(v)\\in[-\\pi\/2,\\pi\/2]$\u4f7f\u5f97$C(v,s(v))$ \u662f$S^3$\u7684\u4e00\u4e2a\u5168\u6d4b\u5730\u7403, \u5373, \u5bf9\u67d0\u4e2a$\\overline{Q}(v)\\in S^3$;<br \/>\n\\[<br \/>\nC(v,s(v))=\\partial B_{\\pi\/2}(\\overline{Q}(v))<br \/>\n\\]<br \/>\n\u4f8b\u5982, \u5982\u679c$v\\in S^3-\\Omega_{3\\varepsilon}$, \u5219 $\\overline{Q}(v)=\\pm v$ \u6216\u8005 $s(v)=\\pm\\pi\/2$, \u5176\u53d6\u503c\u4f9d\u8d56\u4e8e$v$ \u662f\u5c5e\u4e8e $A^{int}$ \u8fd8\u662f $A^{ext}$.<\/li><\/ul>\u4e8b\u5b9e\u4e0a, \u6700\u540e\u4e00\u6761\u4e2d, ${\\overline{Q}(v)}$\u539f\u5219\u4e0a\u53ef\u4ee5\u53d6\u4e24\u4e2a\u503c, \u4f46\u5bf9\u53ef\u5b9a\u5411\u66f2\u9762, \u6211\u4eec\u6709${\\partial B_{\\pi\/2}(p)\\neq \\partial B_{\\pi\/2}(-p)}$, \u56e0\u800c${\\overline{Q}(v)}$\u548c${s(v)}$\u4e00\u6837\u90fd\u662f\u552f\u4e00\u7684.<\/p>\n<p>\u7279\u522b\u5730, \u7531\u4e0a\u9762\u7684\u7b2c\u4e8c\u3001\u4e09\u4e24\u6761, \u6211\u4eec\u6709<br \/>\n\\begin{equation}\\label{eq:2}<br \/>\n\\sup\\limits_{(v,t)\\in\\partial(\\overline{B^4}\\times[-\\pi,\\pi])}\\textrm{area}(C(v,t))=4\\pi<br \/>\n\\end{equation}<br \/>\n\u7a0d\u540e, \u6211\u4eec\u5c06\u770b\u5230\u8fd9\u4e00\u7b49\u5f0f\u6b63\u662f\u4e0a\u9762\u7b2c<a class='latex_ref' href=#sec:2.2>2.2<\/a>\u5c0f\u8282\u4e2d\u8ba8\u8bba\u7684&#8221;\u8fb9\u754c\u969c\u788d&#8221;, \u518d\u6b21\u4e4b\u524d, \u8ba9\u6211\u4eec\u9996\u5148\u5229\u7528${C(v,t)}$\u6765\u5b8c\u6210${\\Sigma\\subset S^3}$\u5bf9\u5e94\u7684\u6781\u5c0f\u6781\u5927\u540c\u4f26\u7c07.<br \/>\n<span class=\"latex_subsection\">2.4.&#x00A0;$\\Sigma\\subset S^3$\u7684\u6781\u5c0f\u6781\u5927\u540c\u4f26\u7c07<a id=\"sec:2.4\"><\/a><\/span>\n<\/p>\n<p>\u5f85\u7eed&#8230;<\/p>\n<div class='bibtex'>\n\t<div class='bibtex_h'>\u53c2\u8003\u6587\u732e<\/div>\n\t<ol><li id='MatthiasErnst2003Existence'><span class='bibtex_author'>M. 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MR<a class='bibtex_mrnumber' target='_blank' href='http:\/\/www.ams.org\/mathscinet-getitem?mr=772125'>772125<\/a><\/li><li id='Lawson1970Complete'><span class='bibtex_author'>J. Lawson<\/span>, <a class='bibtex_title' target='_blank' href='https:\/\/doi.org\/10.2307\/1970625'>Complete minimal surfaces in $Ssp3$<\/a>, <span class='bibtex_journal'>Ann. of Math. (2)<\/span> <span class='bibtex_volume'>92<\/span>(<span class='bibtex_year'>1970<\/span>), <span class='bibtex_page'>335---374<\/span>. MR<a class='bibtex_mrnumber' target='_blank' href='http:\/\/www.ams.org\/mathscinet-getitem?mr=0270280'>0270280<\/a><\/li><li id='LiYau1982new'><span class='bibtex_author'>P. Li and S.  Yau<\/span>, <a class='bibtex_title' target='_blank' href='https:\/\/doi.org\/10.1007\/BF01399507'>A new conformal invariant and its applications to the              Willmore conjecture and the first eigenvalue of compact<\/a>, <span class='bibtex_journal'>Invent. 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Geom.<\/span> <span class='bibtex_volume'>1<\/span>(<span class='bibtex_year'>1993<\/span>), no. <span class='bibtex_number'>2<\/span>, <span class='bibtex_page'>281---326<\/span>. MR<a class='bibtex_mrnumber' target='_blank' href='http:\/\/www.ams.org\/mathscinet-getitem?mr=1243525'>1243525<\/a><\/li><\/ol><\/div>","protected":false},"excerpt":{"rendered":"<p>\u7533\u660e: \u672c\u6587\u8f6c\u81eaMatheus\u2019 Weblog, \u539f\u6587\u9898\u76ee\u4e3aThe Willmore conjecture a&hellip; <a class=\"more-link\" href=\"https:\/\/blog.vanabel.cn\/?p=279\">\u7ee7\u7eed\u9605\u8bfb<span class=\"screen-reader-text\">Fernando Coda Marques\u4e0eAndre Neves \u89e3\u51b3Willmore\u731c\u60f3<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[2],"tags":[49,52,51,53,50],"class_list":["post-279","post","type-post","status-publish","format-standard","hentry","category-math","tag-willmore","tag-tonglun","tag-tuopu","tag-jixiaojida","tag-baopo","entry"],"_links":{"self":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/279","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=279"}],"version-history":[{"count":24,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/279\/revisions"}],"predecessor-version":[{"id":284,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/279\/revisions\/284"}],"wp:attachment":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=279"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=279"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=279"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}