Title: Energy quantization for Willmore surfaces with b… 继续阅读Energy quantization for Willmore surfaces with bounded index
Curvature-Torsion Entropy for Twisted Curves under Curve Shortening Flow
Title: Curvature-Torsion Entropy for Twisted Curves und… 继续阅读Curvature-Torsion Entropy for Twisted Curves under Curve Shortening Flow
A discrete Blaschke Theorem for convex polygons in $2$-dimensional space forms
Title: A discrete Blaschke Theorem for convex polygons … 继续阅读A discrete Blaschke Theorem for convex polygons in $2$-dimensional space forms
A Hilbert Bundles Description of Complex Brunn-Minkowski Theory
Title: A Hilbert Bundles Description of Complex Brunn-M… 继续阅读A Hilbert Bundles Description of Complex Brunn-Minkowski Theory
Title: Classicfication Theory
Authors: Abel Milor
Categories: math.DG math.GR
Comments: Master thesis. 69 pages, 8 figures
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After introducing the simplicial manifolds, such as the different ways of
defining the differential forms on them, we summarized a canonical way of
calculating the characteristic classes of a $G$-principal bundle by computing
them on the classifying bundle $EG\longrightarrow BG$. Finally, we calculated
the first Pontryagin class on the classifying bundle of the Lie matrix groups
and showed that for certain of them, the computed form is equal to the
symplectic form on $BG$ given by some authors up to a constant coefficient.
\\ ( https://arxiv.org/abs/2305.06282 , 442kb)
Title: Some geometric inequalities by the ABP method
Authors: Doanh Pham
Categories: math.DG math.AP
Comments: to appear in International Mathematics Research Notices
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In this paper, we apply the so-called Alexandrov-Bakelman-Pucci (ABP) method
to establish some geometric inequalities. We first prove a logarithmic Sobolev
inequality for closed $n$-dimensional minimal submanifolds $\Sigma$ of $\mathbb
S^{n+m}$. As a consequence, it recovers the classical result that $|\mathbb
S^n| \leq |\Sigma|$ for $m = 1,2$. Next, we prove a Sobolev-type inequality for
positive symmetric two-tensors on smooth domains in $\mathbb R^n$ which was
established by D. Serre when the domain is convex. In the last application of
the ABP method, we formulate and prove an inequality related to
quermassintegrals of closed hypersurfaces of the Euclidean space.
\\ ( https://arxiv.org/abs/2305.05819 , 17kb)