arXiv research

Study finite curvature solutions on surfaces with nonnegative Gauss curvature.

Estimates Bartnik mass for metrics with nonnegative Gauss curvature.

In this paper, we discuss the isometric embedding problem in hyperbolic space with nonnegative extrinsic curvature. We prove a priori bounds for the trace of the second fundamental form H and extend the result to n-dimensions. We also obtain an estimate for the gradient of the smaller principal curvature in 2 dimension…

Study classifies 4D shrinkers with nonnegative Ricci curvature.

We prove the existence of two Alexandrov embedded closed magnetic geodesics on any two dimensional sphere with nonnegative Gauss curvature.

The Gauss-Bonnet inequality holds for certain non-aspherical manifolds up to dimension five.

Study surfaces with nonnegative curvature in spectral sense, proving inequalities and bounds.

In this paper we study self-similar solutions in warped products satisfying $F-\mathcal{F}=\bar{g}(λ(r)\partial_{r},ν)$, where $\mathcal{F}$ is a nonnegative constant and $F$ is in a class of general curvature functions including powers of mean curvature and Gauss curvature. We show that slices are the only closed stri…

Sharp lower bound for curvature in Kähler manifolds.

Unified view of monotonicity formulas for inverse mean curvature flow and $p$-capacitary potentials.

We obtain some nonexistence results for complete noncompact stable hyppersurfaces with nonnegative constant scalar curvature in Euclidean spaces. As a special case we prove that there is no complete noncompact strongly stable hypersurface $M$ in $\mathbb{R}^{4}$ with zero scalar curvature $S_2$, nonzero Gauss-Kronecker…

Paper proves Liouville theorem for curvature equation with boundary conditions.

We study the supremum of the total mean curvature on the boundary of compact, mean-convex 3-manifolds with nonnegative scalar curvature, and a prescribed boundary metric. We establish an additivity property for this supremum and exhibit rigidity for maximizers assuming the supremum is attained. When the boundary consis…

The second H. Weyl curvature invariant of a Riemannian manifold, denoted $h_4$, is the second curvature invariant which appears in the well known tube formula of H. Weyl. It coincides with the Gauss-Bonnet integrand in dimension 4. A crucial property of $h_4$ is that it is nonnegative for Einstein manifolds, hence it p…

In this paper, we obtain several new characterizations of the Clifford torus as a Lagrangian self-shrinker. We first show that the Clifford torus $\mathbb{S}^1(1)\times\mathbb{S}^1(1)$ is the unique compact orientable Lagrangian self-shrinker in $\mathbb{C}^2$ with $|A|^2\leq 2$, which gives an affirmative answer to Ca…

Study on Kähler manifolds with nonnegative Ricci curvature, focusing on rigidity.

Study on rotational hypersurfaces with constant Gauss-Kronecker curvature.

Study rotational surfaces with prescribed Gauss curvature in 3D space.

Paper solves Orlicz-Aleksandrov problem using Gauss curvature flow.

Study Euler characteristic of manifolds with almost nonnegative curvature operator, showing nonnegativity under certain conditions.

The paper extends topological results to noncompact spaces with nonnegative N-Bakry Émery Ricci curvature.

Study open Alexandrov spaces with nonnegative curvature, proving structural results.

Sharp inequalities for manifolds with nonnegative curvature.

Classification of constant curvature surfaces in Berger spheres.

Survey on open manifolds with nonnegative Ricci curvature and open questions.

This note is a study of nonnegativity conditions on curvature which are preserved by the Ricci flow. We focus on specific kinds of curvature conditions which we call noncoercive, these are the conditions for which nonnegative curvature and vanishing scalar curvature doesn't imply flatness. We show that, in dimensions g…

We prove a global smooth isometric immersion for negatively curved surfaces with finite total curvature.

The $(2k)$-th Gauss-Bonnet curvature is a generalization to higher dimensions of the $(2k)$-dimensional Gauss-Bonnet integrand, it coincides with the usual scalar curvature for $k=1$. The Gauss-Bonnet curvatures are used in theoretical physics to describe gravity in higher dimensional space times where they are known a…

Study on Gauss images of specific minimal surfaces with finite curvature.

Minimal hypersurfaces in S^5 with specific curvature properties are totally geodesic.

Study examines noncompact cases of Gauss Curvature Flow on revolution surfaces.

The paper constructs ancient solutions to curvature flows in bounded and unbounded regions.

In this paper, we investigate the Gauss maps of a Ricci-mean curvature flow. A Ricci-mean curvature flow is a coupled equation of a mean curvature flow and a Ricci flow on the ambient manifold. Ruh and Vilms proved that the Gauss map of a minimal submanifold in a Euclidean space is a harmonic map, and Wang extended thi…

We study manifolds with almost nonnegative curvature operator (ANCO) and provide first examples of closed simply connected ANCO mannifolds that do not admit nonnegative curvature operator.

Study relates Gaussian curvature signs to cuspidal edge types and geometric invariants.

New proof of splitting theorem and finite ends of minimal hypersurfaces in nonnegative curvature manifolds.

New theorem links quaternionic-Kähler manifolds to symmetric spaces.

We discuss the cobordism type of spin manifolds with nonnegative sectional curvature. We show that in each dimension $4k \geq 12$, there are infinitely many cobordism types of simply connected and nonnegatively curved spin manifolds. Moreover, we raise and analyze a question about possible cobordism obstructions to non…

Planes and spheres are the only stationary surfaces with constant Gauss curvature.

Study on the regularity of $p$-Gauss curvature flow near flat interfaces.

Study proves rigidity theorems for ancient solutions to mean curvature flow with convex image.

New vanishing theorems for genera derived under almost nonnegative Ricci curvature.

The paper studies CR Yamabe solutions on Sasakian manifolds with nonnegative curvature.

The Gauss-Bonnet curvature of order $2k$ is a generalization to higher dimensions of the Gauss-Bonnet integrand in dimension $2k$, as the usual scalar curvature generalizes the two dimensional Gauss-Bonnet integrand. In this paper, we evaluate the first variation of the integrals of these curvatures seen as functionals…

Using polar convex bodies and the $C_0$-bounds from Guan and Ni \cite{PL}, we obtain a uniform lower bound on the Gauss curvature of the normalized solution of the Gauss curvature flow without using Chow's Harnack inequality \cite{Ch2}.

Ancient Ricci flows with nonnegative curvature operator have bounded entropy.

We show the uniqueness of strictly convex closed smooth self-similar solutions to the $α$-Gauss curvature flow with $(1/n) < α< 1+(1/n)$. We introduce a Pogorelov type computation, and then we apply the strong maximum principle. Our work combined with earlier works on the Gauss Curvature flow imply that the $α$-Gauss c…

We give an estimate of the Gauss curvature for minimal surfaces in ${\mathbb R}^m$ whose Gauss map omits more than $m(m+1)/2$ hyperplanes in ${\mathbb P}^{m-1}({\mathbb C})$.