arXiv research

We extend the Chern-Heinz inequalities about mean curvature and scalar curvature of graphs of $C^{2}$-functions to leaves of transversally oriented codimension one $C^{2}$-foliations of Riemannian manifolds. That extends partially Salavessa's work on mean curvature of graphs and generalize results of Barbosa-Kenmotsu-O…

Sharp curvature bounds for minimal graphs over unit disk.

Estimates mean curvature in warped product spaces, generalizing known results.

Given $(\bar{M},Ω)$ a calibrated Riemannian manifold with a parallel calibration of rank $m$, and $M^m$ an immersed orientable submanifold with parallel mean curvature $H$ we prove that if $\cos θ$ is bounded away from zero, where $θ$ is the $Ω$-angle of $M$, and if $M$ has zero Cheeger constant, then $M$ is minimal. I…

Unified estimates for mean curvature in Lorentz-Minkowski space.

New partial orders on positive-definite matrices derived from geometry.

This is a survey of our work on spacelike graphic submanifolds in pseudo-Riemannian products, namely on Heinz-Chern and Bernstein-Calabi results and on the mean curvature flow, with applications to the homotopy of maps between Riemannian manifolds.

Let $M^n\subset\mathbb R^{n+1}$ be the graph of a $C^2$-real valued function defined in a closed ball of $\mathbb R^n$. In this work, we obtain upper bounds for $\inf_M|H|$ and $\inf_M|R|$, where $H$ and $R$ are, respectively, the mean curvature and the scalar curvature of $M^n$, generalizing estimates given by Heinz i…

In this overview report we generalize Erhard Heinz' curvature estimate for minimal graphs in R^3 to graphs in R^n of prescribed mean curvature. Secondly, we analyse these problems in the frame of the outer differential geometry which leads us to the notions of normal torsion and normal curvature for immersions in R^4.

The paper examines critical points of solutions to a surface equation in 3D spacelike spaces.

Using Schauder's theory for linear elliptic partial differential equations in two independent variables and fundamental estimates for univalent mappings due to E. Heinz we establish an upper bound of the Gaussian curvature of two-dimensional minimal surface graphs in R^n. This leads us to a theorem of Bernstein-Liouvil…

We succeed in writing 2-dimensional conformally invariant non-linear elliptic PDE (harmonic map equation, prescribed mean curvature equations...etc) in divergence form. This divergence free quantities generalize to target manifolds without symmetries the well known conservation laws for harmonic maps into homogeneous s…

Paper estimates Gaussian curvature of minimal graphs in a specific manifold.

Revisits Weyl's problem on isometric immersions of spheres into 3D manifolds.

Survey Bernstein-type theorems for graphical surfaces in Euclidean and Lorentz-Minkowski spaces.

Heinz Hopf's famous fibrations of the 2n+1-sphere by great circles, the 4n+3-sphere by great 3-spheres, and the 15-sphere by great 7-spheres have a number of interesting properties. Besides providing the first examples of homotopically nontrivial maps from one sphere to another sphere of lower dimension, they all share…

The isoperimetric inequality and related inequalities are explored.

New proof of Willmore inequality using geometric divergence inequality.

Lorentz-Finsler geometry reveals new and old inequalities.

The paper derives new inequalities on manifolds and applies them to convex hypersurfaces.

The paper proves inequalities on Finsler manifolds under Ricci curvature bounds.

New inequality on sphere generalizes circle inequality.

Paper proves anisotropic Minkowski inequality and related inequalities.

Explains geometric inequalities for minimal hypersurfaces.

The paper finds new inequalities for convex polygons.

Paper provides an example showing CD inequality doesn't imply CDE' inequality.

The study improves Bochner inequality on Finsler manifolds to derive important inequalities.

The paper proves various inequalities on gradient shrinking Ricci solitons.

Sharp inequality found on three-balls for fourth order Sobolev traces.

Sharp inequalities for star bodies in 2D space.

The paper develops inequalities for log-concave functions and related surface areas.

Sharp inequalities derived from fractional Sobolev inequalities on spheres.

The paper extends matrix inequalities to various types of matrices.

Study on functional inequalities on simple edge spaces.

Proves inequalities on curved spaces with positive curvature.

The paper derives inequalities on Finsler manifolds, influenced by their curvatures.

Paper refines Talagrand inequality on Euclidean spaces.

New inequalities for convex curves with multiple geometric factors.

Alternative proofs for various inequalities on Riemannian manifolds.

Study on Riemannian isoperimetric inequality, finding it true generically.

The paper proves Hardy and Rellich inequalities for submanifolds in Hadamard spaces.

The paper establishes inequalities and gradient estimates for harmonic functions on Finsler measure spaces.

The paper proves a Harnack inequality for heat equations on Finsler metric measure manifolds.

Study functional inequalities on non-reversible Finsler manifolds.

The paper explores how information geometry impacts classical CR inequalities.

Extends Riemannian geometry inequalities with sharper estimates.

Study inequalities on hyperbolic spaces and Riemannian manifolds using symmetrization and heat semigroup.

The paper proves inequalities for hypersurfaces in weighted manifolds.