ArXiv – Page 2 – MathRlife

Van Abel June 1, 2023StatusArXivbubble, Heintze-Karcher inequality, Pohozaev identity, Serrin's type problem Leave a comment

Title A note on Serrin’s type problem on…

Title: A note on Serrin’s type problem on Riemannian manifolds Authors: Allan Freitas, Alberto Roncoroni and M\’arcio Santos Categories: math.DG math.AP Comments: Comments are welcome! \\ In this paper, we deal with Serrin-type problems in Riemannian manifolds. First, we obtain a Heintze-Karcher inequality and a Soap Bubble result, with its respective rigidity, when the ambient space has a Ricci tensor bounded below. After, we approach a Serrin problem in bounded domains of manifolds endowed with a closed conformal vector field. … Continue reading “Title A note on Serrin’s type problem on…”

Van Abel June 1, 2023ArXiv$\Phi_{(3)}$-conservation law, $\phi_3$-harmonic maps, extrinsic average variational method, first variation formula, Liouville type, monotonicity formula, MUSTREAD, second variation formula, sphere theorems, stress-energy tensor, topological vanishing theorems, Unstable manifolds Leave a comment

The geometry of $\Phi_{(3)}$-harmonic maps

Title: The geometry of $\Phi_{(3)}$-harmonic maps Authors: Shuxiang Feng, Yingbo Han, Kaige Jiang and Shihshu Walter Wei Categories: math.DG math-ph math.AP math.MP Comments: 46 pages, to appear in Nonlinear Analysis (2023). arXiv admin note: text overlap with arXiv:1911.05855 MSC-class: 58E20, 53C21, 53C25 \\ In this paper, we motivate and extend the study of harmonic maps or $\Phi_{(1)}$-harmonic maps (cf [15], Remark 1.3 (iii)), $\Phi$-harmonic maps or $\Phi_{(2)}$-harmonic maps (cf. [24], Remark 1.3 (v)), and explore geometric properties of $\Phi_{(3)}$-harmonic maps … Continue reading “The geometry of $\Phi_{(3)}$-harmonic maps”

Van Abel May 30, 2023ArXivharmonic maps, rigidity estimate Leave a comment

Sharp quantitative rigidity results for maps from $S^2$ to $S^2$ of general degree

Title: Sharp quantitative rigidity results for maps from $S^2$ to $S^2$ of general degree Authors: Melanie Rupflin Categories: math.AP math.DG MSC-class: 53C43, 58E20, 30C70, 26D10, \\ As the energy of any map $v$ from $S^2$ to $S^2$ is at least $4\pi \vert deg(v)\vert$ with equality if and only if $v$ is a rational map one might ask whether maps with small energy defect $\delta_v=E(v)-4\pi \vert deg(v)\vert$ are necessarily close to a rational map. While such a rigidity statement turns out … Continue reading “Sharp quantitative rigidity results for maps from $S^2$ to $S^2$ of general degree”

Van Abel May 30, 2023ArXivHiggs, Hitchin, hyperkähler Leave a comment

Asymptotic behaviour of the Hitchin metric on the moduli space of Higgs bundles

Asymptotic behaviour of the Hitchin metric on the moduli space of Higgs bundles Takuro Mochizuki The moduli space of stable Higgs bundles of degree 0 is equipped with the hyperkähler metric, called the Hitchin metric. On the locus where the Hitchin fibration is smooth, there is the hyperkähler metric called the semi-flat metric, associated with the algebraic integrable systems with the Hitchin section. We prove the exponentially rapid decay of the difference between the Hitchin metric and the semi-flat metric … Continue reading “Asymptotic behaviour of the Hitchin metric on the moduli space of Higgs bundles”

Van Abel May 23, 2023QuoteArXivChen-Yau, divergence, eigenvalue, inequality, Laplacian, Riemannian, Yang-type first inequality Leave a comment

Inequalities for eigenvalues of operators in divergence form on Riemannian manifolds isometrically immersed in Euclidean space

Authors: Cristiano Silva, Juliana Miranda, Marcio Ara\’ujo Filho Categories: math.DG math.SP Comments: 15 pages \\ In this paper, we compute universal inequalities of eigenvalues of a large class of second-order elliptic differential operators in divergence form, that includes, e.g., the Laplace and Cheng-Yau operators, on a bounded domain in a complete Riemannian manifolds isometrically immersed in Euclidean space. A key step in order to obtain the sequence of our estimates is to get the right Yang-type first inequality. We also … Continue reading “Inequalities for eigenvalues of operators in divergence form on Riemannian manifolds isometrically immersed in Euclidean space”

Van Abel May 16, 2023ArXivconformal Gauss map, De Sitter space Leave a comment

Energy quantization for Willmore surfaces with bounded index

Title: Energy quantization for Willmore surfaces with bounded index Authors: Dorian Martino Categories: math.DG math.AP Comments: 49 pages \\ We prove an energy quantization result for Willmore surfaces with bounded index, whether the underlying Riemann surfaces degenerates in the moduli space or not. To do so, we translate the question on the conformal Gauss map’s point of view. In particular, we prove that in a neck or a collar region, the conformal Gauss map converges to a light-like geodesic in … Continue reading “Energy quantization for Willmore surfaces with bounded index”