Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 0707.2056

Integral formulas for a class of curvature PDE's and applications to isoperimetric inequalities and to symmetry problems

Vittorio Martino·Annamaria Montanari
View original source ↗Read source document ↗Export metadata ↓

Abstract

We prove integral formulas for closed hypersurfaces in C^{n+1}, which furnish a relation between elementary symmetric functions in the eigenvalues of the complex Hessian matrix of the defining function and the Levi curvatures of the hypersurface. Then we follow the Reilly approach to prove an isoperimetric inequality. As an application, we obtain the ``Soap Bubble Theorem'' for star-shaped domains with positive and constant Levi curvatures bounding the classical mean curvature from above.

Explore related subjects

math.APmath.CV

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Vittorio Martino, Annamaria Montanari. 2007-07-13. Integral formulas for a class of curvature PDE's and applications to isoperimetric inequalities and to symmetry problems. https://arxiv.org/abs/0707.2056

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Smooth solutions to the chord log-Minkowski problem

In integral geometry generalized with Aleksandrov's variational theory, Lutwak-Xi-Yang-Zhang [Comm. Pure Appl. Math. 77 (2024)] recently opened the door to researching the cone-chord measures and their log-Minkowski problem stemming from the chord integrals, named as the chord log-Minkowski problem. In the smooth category, the solvability of the chord log-Minkowski problem amounts to dealing with a nonlocal {M}onge-{A}mpère equation involving a Riesz potential. In this paper, to study the chord log-Minkowski problem, we first present some novel results on the boundary regularity of the Riesz potential. Based on these results, we obtain the regularity and existence for the chord log-Minkowski problem from the perspective of a nonlocal {M}onge-{A}mpère equation and a nonlocal Gauss curvature flow equation.

math.AP↗
arXiv2026

A proof of Onsager's Conjecture for the SQG equation

We construct solutions to the SQG equation that fail to conserve the Hamiltonian while having the maximal allowable regularity for this property to hold. This result solves the generalized Onsager conjecture on the threshold regularity for Hamiltonian conservation for SQG. We also prove nonuniqueness for arbitrary smooth data and an h-principle result.

math.AP↗