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Shanze Gao

Publications and source records attributed to Shanze Gao.

11 recordsLinked to original sources
arXiv2026

Hopf type theorem for surfaces of constant weighted mean curvature in Riemannian manifolds

We consider compact immersed surfaces of genus zero in a three-dimensional smooth metric measure space \((M^3,g,e^{-f}\mathrm{d}V_{g})\). We introduce a new weighted mean curvature and prove that any such surface whose weighted mean curvature is constant must be totally umbilical, provided that a condition relating the Ricci curvature of $M$ and the weight function $f$ is satisfied. We also give an application of this result to the uniqueness of the dual Christoffel-Minkowski problem.

math.DG↗
arXiv2026

Rigidity of spacelike hypersurface with constant curvature and intersection angle condition

In the Minkowski space, we consider a compact, spacelike hypersurface with boundary, which can be written as a graph on a spacelike hyperplane. We prove that, if its $k$-th mean curvature is constant, and its boundary is on the hyperplane with constant intersection angles, then the hypersurface must be a part of a hyperboloid, unless it is entirely contained in the hyperplane. The proof is based on an auxiliary function and associated integral equality.

math.DG↗
arXiv2025

Convex spacelike hypersurface of constant curvature with boundary on a hyperboloid

We consider convex, spacelike hypersurfaces with boundaries on some hyperboloid (or lightcone) in the Minkowski space. If the hypersurface has constant higher order mean curvature, and the angle between the normal vectors of the hypersurface and the hyperboloid (or the lightcone) is constant on the boundary, then the hypersurface must be a part of another hyperboloid.

math.DG↗
arXiv2023

Overdetermined problems for fully nonlinear equations with constant Dirichlet boundary conditions in space forms

We consider overdetermined problems for two classes of fully nonlinear equations with constant Dirichlet boundary conditions in a bounded domain in space forms. We prove that if the domain is star-shaped, then the solution to the Hessian quotient overdetermined problem is radially symmetric. By establishing a Rellich-Pohožaev type identity for the $k$-Hessian equation with constant Dirichlet boundary condition, we also show the radial symmetry of the solution to the $k$-Hessian overdetermined problem for some boundary value without star-shapedness assumption of the domain.

math.AP↗
arXiv2023

Closed self-similar solutions to flows by negative powers of curvature

In some warped product manifolds including space forms, we consider closed self-similar solutions to curvature flows whose speeds are negative powers of mean curvature, Gauss curvature and other curvature functions with suitable properties. We prove such self-similar solutions, not necessarily strictly convex for some cases, must be slices of warped product manifolds. A new auxiliary function is the key of the proofs.

math.DG↗
arXiv2021

Self-similar solutions to fully nonlinear curvature flows by high powers of curvature

In this paper, we investigate closed strictly convex hypersurfaces in $\mathbb{R}^{n+1}$ which shrink self-similarly under a large family of fully nonlinear curvature flows by high powers of curvature. When the speed function is given by powers of a homogeneous of degree $1$ and inverse concave function of the principal curvatures with power greater than $1$, we prove that the only such hypersurfaces are round spheres. We also prove that slices are the only closed strictly convex self-similar solutions to such curvature flows in the hemisphere $\mathbb{S}^{n+1}_{+}$ with power greater than or equal to $1$.

math.DG↗
arXiv2019

Characterizations of umbilic hypersurfaces in warped product manifolds

We consider closed orientable hypersurfaces in a wide class of warped product manifolds, which include space forms, deSitter-Schwarzschild and Reissner-Nordström manifolds. By using a new integral formula or Brendle's Heintze-Karcher type inequality, we present some new characterizations of umbilic hypersurfaces. These results can be viewed as generalizations of the classical Jellet-Liebmann theorem and the Alexandrov theorem in Euclidean space.

math.DG↗
arXiv2018

Self-similar solutions of curvature flows in warped products

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 strictly convex self-similar solutions in the hemisphere for such $F$. We also obtain a similar uniqueness result in hyperbolic space $\mathbb{H}^{3}$ for Gauss curvature $F$ and $\mathcal{F}\geq 1$.

math.DG↗
arXiv2017

Uniqueness of closed self-similar solutions to $σ_k^α$-curvature flow

By adapting the test functions introduced by Choi-Daskaspoulos \cite{c-d} and Brendle-Choi-Daskaspoulos \cite{b-c-d} and exploring properties of the $k$-th elementary symmetric functions $σ_{k}$ intensively, we show that for any fixed $k$ with $1\leq k\leq n-1$, any strictly convex closed hypersurface in $\mathbb{R}^{n+1}$ satisfying $σ_{k}^α=\langle X,ν\rangle$, with $α\geq \frac{1}{k}$, must be a round sphere. In fact, we prove a uniqueness result for any strictly convex closed hypersurface in $\mathbb{R}^{n+1}$ satisfying $F+C=\langle X,ν\rangle$, where $F$ is a positive homogeneous smooth symmetric function of the principal curvatures and $C$ is a constant.

math.DG↗
arXiv2016

Self-similar solutions of $σ_k^α$-curvature flow

In this paper, employing a new inequality, we show that under certain curvature pinching condition, the strictly convex closed smooth self-similar solution of $σ_k^α$-flow must be a round sphere. We also obtain a similar result for the solutions of $F=-\langle X, e_{n+1}\rangle \, (*)$ with a non-homogeneous function $F$. At last, we prove that if $F$ can be compared with $\frac{(n-k+1)σ_{k-1}}{kσ_{k}}$, then a closed strictly $k$-convex solution of $(*)$ must be a round sphere.

math.DG↗