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Weimin Sheng

Publications and source records attributed to Weimin Sheng.

At least 19 recordsLinked to original sources
arXiv2025

Noncollapsing for Curvature Flows with Inhomogeneous Speeds

We study closed, embedded hypersurfaces in Euclidean space evolving by fully nonlinear curvature flows, whose speed is given by a symmetric, monotone increasing, $1$-homogeneous, positive underlying speed function $F$ composed with a modulating function $\Psi$. Under the assumption that $F$ is convex or inverse-concave and that $\Psi$ satisfies the corresponding structural conditions, we establish exterior noncollapsing estimates for the flow. The main difficulty stems from the nonlinearity of the evolution equation satisfied in the viscosity sense by the exscribed curvature, whereas in previous works it is a solution to the linearized flow. Moreover, in the case where $F$ is inverse-concave, we refine Andrews and Langford's argument for the interior case.

math.DG↗
arXiv2025

Long time behavior of a class of non-homogeneous anisotropic fully nonlinear curvature flows

In this paper, we study a class of non-homogeneous anisotropic fully nonlinear curvature flows in $\mathbb{R}^{n+1}$. More precisely, we consider a hypersurface $M$ in $\mathbb{R}^{n+1}$ deformed by a flow along its unit normal with its speed $f(r)\sigma_k^\alpha$ where $\sigma_k$ is the $k$-th elementary symmetric polynomial of $M$'s principle curvatures, $r$ is the distance of the point on $M$ to the origin, $f$ is a smooth nonnegative function on $[0,\infty)$ and $\alpha > 0$. Under some suitable conditions on $f$, we prove that starting from a star-shaped and $k$-convex hypersurface, the flow exists for all time and converges smoothly to a sphere after normalization. In particular, we generalize the results in \cite{li2022asymptotic}.

math.DG↗
arXiv2025

On the new weighted geometric inequalities near the sphere in space forms

In this paper, we first investigate weighted Minkowski type inequalities for nearly spherical sets in space forms, focusing on the sets that are $C^1$-close to geodesic spheres. Our results generalize the work of \cite{G22} by incorporating broader geometric settings and convex weight functions. Additionally, we establish quantitative stability estimates for weighted Alexandrov-Fenchel type inequalities in $\mathbb{R}^{n+1}$ and $\mathbb{H}^{n+1}$, extending the earlier results of \cite{VW24} and \cite{ZZ23}. These inequalities hold for nearly spherical sets that are $W^{2,\infty}$-close to geodesic spheres coupled with general convex weights.

math.DG↗
arXiv2025

On rigidity of hypersurfaces with constant shifted curvature functions in warped product manifolds

In this paper, we give some new characterizations of umbilic hypersurfaces in general warped product manifolds, which can be viewed as generalizations of the work in \cite{KLP18} and \cite{WX14}. Firstly, we prove the rigidity for hypersurfaces with constant linear combinations of shifted higher order mean curvatures. Using integral inequalities and Minkowski-type formulas, we then derive rigidity theorems in sub-static warped product manifolds, including cases that the hypersurface satisfies some nonlinear curvature conditions. Finally, we show that our results can be applied to more general warped product manifolds, including the cases with non-constant sectional curvature fiber.

math.DG↗
arXiv2024

On rigidity of hypersurfaces with constant shifted curvature functions in hyperbolic space

In this paper, we first give some new characterizations of geodesic spheres in the hyperbolic space by the condition that hypersurface has constant weighted shifted mean curvatures, or constant weighted shifted mean curvature ratio, which generalize the result of Hu-Wei-Zhou \cite{HWZ23}. Secondly, we investigate several rigidity problems for hypersurfaces in the hyperbolic space with constant linear combinations of weighted shifted mean curvatures as well as radially symmetric shifted mean curvatures. As applications, we obtain the rigidity results for hypersurfaces with constant linear combinations of mean curvatures in a general form and constant Gauss-Bonnet curvature $L_k$ under weaker conditions, which extend the work of the third author and Xia \cite{WX14}.

math.DG↗
arXiv2022

Flow by Gauss curvature to the $L_p$-Gaussian Minkowski problem

In this paper, we study the $L_p$-Gaussian Minkowski problem, which arises in the $L_p$-Brunn-Minkowski theory in Gaussian probability space. We use Aleksandrov's variational method with Lagrange multipliers to prove the existence of the logarithmic Gauss Minkowski problem. We construct a suitable Gauss curvature flow of closed, convex hypersurfaces in the Euclidean space $\mathbb{R}^{n+1}$, and prove its long-time existence and converges smoothly to a smooth solution of the normalized $L_p$ Gaussian Minkowski problem in cases of $p>0$ and $-n-1<p\leq 0$ with even prescribed function respectively. We also provide a parabolic proof in the smooth category to the $L_p$-Gaussian Minkowski problem in cases of $p\geq n+1$ and $0<p<n+1$ with even prescribed function, respectively.

math.DG↗
arXiv2022

A flow approach to the Musielak-Orlicz-Gauss image problem

In this paper, the extended Musielak-Orlicz-Gauss image problem is studied. Such a problem aims to characterize the Musielak-Orlicz-Gauss image measure $\widetilde{C}_{G,Ψ,λ}(Ω,\cdot)$ of convex body $Ω$ in $\mathbb{R}^{n+1}$ containing the origin (but the origin is not necessary in its interior). In particular, we provide solutions to the extended Musielak-Orlicz-Gauss image problem based on the study of suitably designed parabolic flows, and by the use of approximation technique (for general measures). Our parabolic flows involve two Musielak-Orlicz functions and hence contain many well-studied curvature flows related to Minkowski type problems as special cases. Our results not only generalize many previously known solutions to the Minkowski type and Gauss image problems, but also provide solutions to those problems in many unsolved cases.

math.DG↗
arXiv2021

Weak scalar curvature lower bounds along Ricci flow

In this paper, we study Ricci flow on compact manifolds with a continuous initial metric. It was known from Simon that the Ricci flow exists for a short time. We prove that the scalar curvature lower bound is preserved along the Ricci flow if the initial metric has a scalar curvature lower bound in distributional sense provided that the initial metric is $W^{1,p}$ for some $n<p\le \infty$. As an application, we use this result to study the relation between Yamabe invariant and Ricci flat metrics. We prove that if the Yamabe invariant is nonpositive and the scalar curvature is nonnegative in distributional sense, then the manifold is isometric to a Ricci flat manifold.

math.DG↗
arXiv2020

Removable singularity of positive mass theorem with continuous metrics

In this paper, we consider asymptotically flat Riemannnian manifolds $(M^n,g)$ with $C^0$ metric $g$ and $g$ is smooth away from a closed bounded subset $Σ$ and the scalar curvature $R_g\ge 0$ on $M\setminus Σ$. For given $n\le p\le \infty$, if $g\in C^0\cap W^{1,p}$ and the Hausdorff measure $\mathcal{H}^{n-\frac{p}{p-1}}(Σ)<\infty$ when $n\le p<\infty$ or $\mathcal{H}^{n-1}(Σ)=0$ when $p=\infty$, then we prove that the ADM mass of each end is nonnegative. Furthermore, if the ADM mass of some end is zero, then we prove that $(M^n,g)$ is isometric to the Euclidean space by showing the manifold has nonnegative Ricci curvature in RCD sense. This extends the result of [Lee-LeFloch2015] from spin to non-spin, also improves the result of [Shi-Tam2018] and [Lee2013]. Moreover, for $p=\infty$, this confirms a conjecture of Lee [Lee2013].

math.DG↗
arXiv2020

An Anisotropic shrinking flow and L_p Minkowski problem

We consider a shrinking flow of smooth, closed, uniformly convex hypersurfaces in (n+1)-dimensional Euclidean space with speed fu^{alpha}{sigma}_n^{beta}, where u is the support function of the hypersurface, alpha, beta are two constants, and beta>0, sigma_n is the n-th symmetric polynomial of the principle curvature radii of the hypersurface. We prove that the flow has a unique smooth and uniformly convex solution for all time, and converges smoothly after normalisation, to a soliton which is a solution of an elliptic equation, when the constants alpha, beta belong to a suitable range, provided the initial hypersuface is origin-symmetric and f is a smooth positive even function on S^n. For the case alpha>= 1+n*beta, beta>0, we prove that the flow converges smoothly after normalisation to a unique smooth solution of an elliptic equation without any constraint on the initial hypersuface and smooth positive function f. When beta=1, our argument provides a uniform proof to the existence of the solutions to the equation of L_p Minkowski problem for p belongs to (-n-1,+infty).

math.DG↗
arXiv2019

A class of anisotropic expanding curvature flows

We consider an expanding flow of smooth, closed, uniformly convex hypersurfaces in (n+1)-dimensional Euclidean space with speed fu^{alpha}{sigma}_k^{beta}, where u is the support function of the hypersurface, alpha, beta are two constants, and beta>0, sigma_k is the k-th symmetric polynomial of the principle curvature radii of the hypersurface, k is an integer and 1<= k<= n. We prove that the flow has a unique smooth and uniformly convex solution for all time, and converges smoothly after normalisation, to a soliton which is a solution of a elliptic equation, when the constants alpha, beta belong to a suitable range, and the function f satisfies a strictly spherical convexity condition. When beta=1, the soliton equation is just the equation of Lp Christoffel-Minkowski problem. Thus our argument provides a proof to the well-known L_p Christoffel-Minkowski problem for the case p>= k+1 where p=2-alpha, which is identify with Ivaki's recent result.

math.DG↗
arXiv2018

On existence of the prescribing $k$-curvature of the Einstein tensor

In this paper, we study the problem of conformally deforming a metric on a $3$-dimensional manifold $M^3$ such that its $k$-curvature equals to a prescribed function, where the $k$-curvature is defined by the $k$-th elementary symmetric function of the eigenvalues of the Einstein tensor, $1\le k\le 3$. We prove the solvability of the problem and the compactness of the solution sets on manifolds when $k=2$ and $3$, provided the conformal class admits a negative $k$-admissible metric with respect to the Einstein tensor.

math.DG↗
arXiv2018

The Exponential Convergence of the CR Yamabe Flow

In this paper, we study the CR Yamabe flow with zero CR Yamabe invariant. We use the CR Poincaré inequality and a Gagliardo-Nirenberg type interpolation inequality to show that this flow has long time solution and the solution converges to a contact form with flat pseudo-Hermitian scalar curvature exponentially.

math.DG↗
arXiv2018

On Variational Properties of Quadratic Curvature Functionals

In this paper, we investigate a class of quadratic Riemannian curvature functionals on closed smooth manifold $M$ of dimension $n\ge 3$ on the space of Riemannian metrics on $M$ with unit volume. We study the stability of these functionals at the metric with constant sectional curvature as its critical point.

math.DG↗
arXiv2017

Flow by Gauss curvature to the Aleksandrov and dual Minkowski problems

In this paper we study a contracting flow of closed, convex hypersurfaces in the Euclidean space $\mathbb R^{n+1}$ with speed $f r^α K$, where $K$ is the Gauss curvature, $r$ is the distance from the hypersurface to the origin, and $f$ is a positive and smooth function. If $α\ge n+1$, we prove that the flow exists for all time and converges smoothly after normalisation to a soliton, which is a sphere centred at the origin if $f \equiv 1$. Our argument provides a parabolic proof in the smooth category for the classical Aleksandrov problem, and resolves the dual q-Minkowski problem introduced by Huang, Lutwak, Yang and Zhang (Acta Math. 216 (2016): 325-388), for the case $q<0$. If $α< n+1$, corresponding to the case $q>0$, we also establish the same results for even function $f$ and origin-symmetric initial condition, but for non-symmetric $f$, counterexample is given for the above smooth convergence.

math.AP↗
arXiv2017

Convergence of the CR Yamabe Flow

We consider the CR Yamabe flow on a compact strictly pseudoconvex CR manifold $M$ of real dimension $2n+1$. We prove convergence of the CR Yamabe flow when $n=1$ or $M$ is spherical.

math.DG↗
arXiv2014

Evolving hypersurfaces by their mean curvature in the background manifold evolving by Ricci flow

We consider the problem of deforming a one-parameter family of hypersurfaces immersed into closed Riemannian manifolds with positive curvature operator. The hypersurface in this family satisfies mean curvature flow while the ambient metric satisfying the normalized Ricci flow. We prove that if the initial metric of the background manifold is sufficiently pinched and the initial hypersurface also satisfies a suitable pinching condition, then either the hypersurfaces shrink to a round point in finite time or converge to a totally geodesic sphere as the time tends to infinity.

math.DG↗