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Gaoming Wang

Publications and source records attributed to Gaoming Wang.

17 recordsLinked to original sources
arXiv2026

Mixed Radial Volume Comparison under Spectral Ricci Bounds

Let $(M^n,g)$ be complete, let $u>0$, and assume $\operatorname{Ric}_g-\alpha\frac{\Delta_g u}{u}g\ge (n-1)\kappa g$. We introduce mixed radial balls associated with the conformal metric $u^{2\alpha}g$. For these mixed radial balls, we obtain a space-form-sharp model comparison and polynomial weighted volume growth without pointwise bounds on $u$. As applications, we give a radial derivation of the spectral Bonnet--Myers and sharp volume theorem of Antonelli--Xu, and a short volume growth derivation of the stable Bernstein theorem in $\mathbb{R}^4$.

math.DG↗
arXiv2026

Stable Minimal Hypersurfaces in Positively Curved $4$-Manifolds

Let $M^3\to X^4$ be a complete, connected, two-sided stable minimal immersion. We prove that if the ambient sectional curvature is nonnegative and the ambient scalar curvature has a positive uniform lower bound, then $M$ is totally geodesic and its normal Ricci curvature vanishes. No weak bounded geometry assumption and no upper curvature bound are imposed. We also construct a complete metric of strictly positive sectional curvature on $\mathbb{R}^4$ admitting a complete, embedded, one-ended, nonparabolic, two-sided stable minimal hypersurface diffeomorphic to $\mathbb{R}^3$ which is not totally geodesic. The rigidity proof combines spectral splitting theory, a warped $\mu$-bubble construction, and a harmonic function level set argument. The example is obtained by a compactly supported deformation of an example in \cite{CLS}.

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arXiv2026

Regularity of branched stable minimal immersed hypersurfaces

We establish a sharp bound on the Hausdorff dimension of the non-branch singular set of branched stable minimal immersed hypersurfaces whose singular sets have locally finite $\mathcal H^{n-2}$-measure: the non-branch singular set is empty when $n=2$, discrete when $n=3$, and has Hausdorff dimension at most $n-3$ when $n\geq4$. We also construct a non-flat stable minimal cone in $\mathbb R^4$ arising from a branched minimal immersion whose vertex is a non-branch singularity. Taking products with Euclidean factors yields examples whose non-branch singular sets have Hausdorff dimension exactly $n-3$, showing that our regularity bound is sharp in every dimension $n\geq3$. The main ingredients in our proof are a generalized Schoen inequality and a corresponding branched sheeting theorem near stationary classical cones and unions of hyperplanes.

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arXiv2026

Regularity of stable capillary minimal hypersurfaces

We develop a regularity and compactness theory for stable capillary minimal hypersurfaces in the half-space $\mathbb{H}^{n+1}$ with contact angle $\theta \in (0,\pi)$ and dimension $n \geq 2$. One key analytic ingredient is a capillary differential Schoen inequality, which allows us to establish a boundary sheeting theorem in the spirit of Bellettini. The other ingredient is a refined classification of stable capillary minimal cones, and we show that for any contact angle $\theta\in(0,\pi)$, the stable capillary minimal hypercone in $\mathbb H^5$ with an isolated singularity must be flat. As a consequence, for any $n\leq4$ and $\theta\in(0,\pi)$, we obtain the Bernstein theorem for embedded complete stable capillary minimal hypersurfaces in $\mathbb H^{n+1}$ with Euclidean area growth.

math.DG↗
arXiv2026

Spectral splitting theorem and ends of minimal hypersurfaces

In this paper, we give a new proof of the splitting theorem on manifolds with nonnegative spectral Ricci curvature proved in [APX24, CMMR24, HW26]. Furthermore, by constructing weighted minimizing geodesics at infinity, we show that minimal hypersurfaces with finite index in manifolds with nonnegative biRic curvature must have finite ends, generalizing the result of Li-Wang [LW04] on manifolds with nonnegative sectional curvature.

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arXiv2025

A splitting theorem for manifolds with spectral nonnegative Ricci curvature and mean-convex boundary

We prove a splitting theorem for a smooth noncompact manifold with (possibly noncompact) boundary. We show that if a noncompact manifold of dimension $n\geq 2$ has $\lambda_1(-\alpha\Delta+\operatorname{Ric})\geq 0$ for some $\alpha<\frac{4}{n-1}$ and mean-convex boundary, then it is either isometric to $\Sigma\times \mathbb{R}_{\geq 0}$ for a closed manifold $\Sigma$ with nonnegative Ricci curvature or it has no interior ends.

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arXiv2025

Scalar curvature rigidity of domains in a 3-dimensional warped product

A warped product with a spherical factor and a logarithmically concave warping function satisfies a scalar curvature rigidity of the Llarull type. We develop a scalar curvature rigidity of the Llarull type for a general class of domains in a three dimensional spherical warped product. In the presence of rotational symmetry, we identify this class of domains as those satisfying a boundary condition analogous to the logarithmic concavity of the warping function.

math.DG↗
arXiv2025

A splitting theorem for 3-manifold with nonnegative scalar curvature and mean-convex boundary

We show that a Riemannian 3-manifold with nonnegative scalar curvature and mean-convex boundary is flat if it contains an absolutely area-minimizing (in the free boundary sense) half-cylinder or strip. Analogous results also hold for a $\theta$-energy-minimizing half-cylinder, or, under certain topological assumptions, a $\theta$-energy-minimizing strip for $\theta\in (0,\pi)$.

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arXiv2024

On $\delta$-Stable Minimal Hypersurfaces in $\mathbb{R}^{n+1}$

In this paper, we extend several results established for stable minimal hypersurfaces to $\delta$-stable minimal hypersurfaces. These include the regularity and compactness theorems for immersed $\delta$-stable minimal hypersurfaces in $\mathbb{R}^{n+1}$ when $n \geq 3$ and $\delta > \frac{n-2}{n}$, as well as the $\delta$-stable Bernstein theorem for $n=3$ and $n=4$ for properly immersion. The range of $\delta$ is optimal, as the $n$-dimensional catenoid in $\mathbb{R}^{n+1}$ is $\frac{n-2}{n}$-stable.

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arXiv2024

Allard-Type Regularity for Varifolds with Prescribed Contact Angle

Given a bounded $C^2$ domain in $\mathbb{R}^{n+1}$ and an integral $n$-rectifiable varifold $V$ with bounded first variation and bounded generalized mean curvature. Given a $C^1$ function $θ$ defined on the boundary of the domain with range $(0,π)$, we assume $V$ has prescribed contact angle $θ$ with $\partial Ω$ and the tangent cone of $V$ at a point $X \in \partial Ω$ is a half-hyperplane of density one. Then we can show that the support of $V$ is a $C^{1,γ}$ hypersurface with boundary near $X$ for some $γ\in (0,1)$.

math.DG↗
arXiv2023

Scalar curvature comparison of rotationally symmetric sets

Let $(M, g)$ be a compact 3-manifold with nonnegative scalar curvature $R_g\geq 0$. The boundary $\partial M$ is diffeomorphic to the boundary of a rotationally symmetric and weakly convex body $\bar{M}$ in $\mathbb{R}^3$. We call $(\bar{M}, \delta)$ a model or a reference. Let $H_{\partial M}$ and $\bar{H}_{\partial M}$ be respectively the mean curvatures of $\partial M$ in $(M, g)$ and $\partial M$ in $(\bar{M}, \delta)$, $\sigma$ and $\bar{\sigma}$ be the induced metric from $g$ and $\delta$. We show that for some classes of $\partial M$, if $H_{\partial M} \geq \bar{H}_{\partial M}$, $\sigma \geq \bar{\sigma}$ and the dihedral angles at the nonsmooth part of $\partial M$ are no greater than the model, then $M$ is flat. We also generalize this result to the hyperbolic case and some spaces with $\mathbb{S}^1$-symmetry. Our approach is inspired by Gromov.

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arXiv2022

Generalized Bernstein Theorem for Stable Minimal Plateau Surfaces

In this paper, we consider a Generalized Bernstein Theorem for a type of generalized minimal surfaces, namely minimal Plateau surfaces. We show that if an orientable minimal Plateau surface is stable and has quadratic area growth in $\mathbb{R}^3 $, then it is flat.

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arXiv2022

Dihedral rigidity in hyperbolic 3-space

We prove a comparison theorem for certain types of polyhedra in a 3-manifold with its scalar curvature bounded below by $-6$. The result confirms in some cases the Gromov dihedral rigidity conjecture in hyperbolic $3$-space.

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arXiv2021

Index of Embedded Networks in the Sphere

In this paper, we will compute the Morse index and nullity for the stationary embedded networks in spheres. The key theorem in the computation is that the index (and nullity) for the whole network is related to the index (and nullity) of small networks and the Dirichlet-to-Neumann map defined in this paper. Finally, we will show that for all stationary triple junction networks in $\mathbb{S}^2$, there is only one eigenvalue (without multiplicity) $-1$, which is less than 0, and the corresponding eigenfunctions are locally constant. Besides, the multiplicity of eigenvalues 0 is 3 for these networks, and their eigenfunctions are generated by the rotations on the sphere.

math.DG↗
arXiv2021

Curvature estimates for stable minimal surfaces with a common free boundary

The minimal surfaces meeting in triples with equal angles along a common boundary naturally arise from soap films and other physical phenomenon. They are also the natural extension of the usual minimal surface. In this paper, we consider the multiple junction surface and show the Bernstein Theorem still holds for stable multiple junction surface in some special case. The key part is to derive the $L^p$ estimate of the curvature for multiple junction surface.

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