arXiv subjects
Publications and source records attributed to Shiguang Ma.
We establish a refined singularity estimate for nonnegative n-superharmonic functions. For complete noncompact locally conformally flat manifolds with nonnegative Ricci curvature, we analytically characterize the volume growth, verify Yau's conjecture on the Cohn-Vossen inequality, and prove a sharp gap theorem that removes all auxiliary assumptions from earlier works.
In this paper we extend the research on potential theory and its geometric applications from Euclidean spaces to homogeneous Carnot groups. We introduce a new approach to use the geometric completeness to estimate the Hausdorff dimension of polar sets of potentials of nonnegative Radon measures for sub-Laplacians in homogeneous Carnot groups. Our approach relies on inequalities that are analogous to the classic integral inequalities about Riesz potentials in Euclidean spaces. Our approach also uses extensions of some of geometric measure theory to homogeneous Carnot groups and the polar coordinates with horizontal radial curves constructed by Balogh and Tyson for polarizable Carnot groups. As consequences, we develop applications of potentials for sub-Laplacians in CR geometry, quaternionic CR geometry, and octonionic CR geometry.
In this paper, we want to give an exposition of our recent work on linear and nonlinear potential theory and their applications in conformal geometry. We use potential theory to study linear and quasilinear equations arising from conformal geometry. We establish the asymptotic behavior near singularities and derive applications in conformal geometry. In particular, we establish some Huber's type theorems and Hausdorff dimension estimates of the ends in conformal geometry in general dimensions.
We provide a complete classification of the asymptotic behavior of isolated singularities for solutions satisfying \[ 0\le-\Delta_{p}u(x)\le \tau u^{\frac{n(p-1)}{n-p}} (x),\,\,u(x)\ge0,\,\,1<p<n,\,\,n\ge2, \]where $u(x)\in C^2(B(0,1)\backslash\{0\})$ and $B(0,1)\subset \mathbb R^n$.
In this article, we will prove existence results for the equations of the type $-Δ_{N}u=H_{l}(u)+μ$ and $F_{\frac{N}{2}}[-u]=H_{l}(u)+μ$ in a bounded domain $Ω$, with Dirichlet boundary condition, where the source term $H_{l}(r)$ takes the form $e^{r}-\sum_{j=0}^{l-1}\frac{r^{j}}{j!}$ and $μ$ is a nonnegative Radon measure.
In this paper we develop the p-thinness and the p-fine topology for the asymptotic behavior of p-superharmonic functions at singular points. We consider these as extensions of earlier works on superharmonic functions in dimension 2, on the Riesz and Log potentials in higher dimensions,, and on p-harmonic functions. It is remarkable that, contrary to the above cases, the p-thinness for the singular behavior differs from the p-thinness for continuity by the Wiener criterion for p-superharmonic functions. As applications of asymptotic estimates of p-superharmonic functions, we also obtain asymptotic estimates of solutions to a class of fully nonlinear elliptic equations. This paper grows out of our recent papers on the potential theory in conformal geometry.
In this paper we introduce the p-Laplace equations for the intermediate Schouten curvature in conformal geometry. These p-Laplace equations provide more tools for the study of geometry and topology of manifolds. First, the positivity of the intermediate Schouten curvature yields the vanishing of Betti numbers on locally conformally flat manifolds as consequences of the Böchner formula as in the works of Nayatani and Guan-Lin-Wang. Secondly and more interestingly, when the intermediate Schouten curvature is nonnegative, these p-Laplace equations facilitate the geometric applications of p-superharmonic functions and the nonlinear potential theory. This leads to the estimates on Hausdorff dimension of singular sets and vanishing of homotopy groups that is inspired by and extends the work of Schoen-Yau. In the forthcoming paper we will present our results on the asymptotic behavior of p-superharmonic functions at singularities.
In this paper we derive estimates for linear potentials that hold away from thin subsets. And, inspired by the celebrated work of Huber, we verify that, for a subset that is thin at a point, there is always a geodesic that reaches to the point and avoids the thin subset in general dimensions. As applications of these estimates on linear potentials, we consider the scalar curvature equations and slightly improve the results of Schoen-Yau and Carron on the Hausdorff dimensions of singular sets which represent the ends of complete conformal metrics on domains in manifolds of dimensions greater than 3. We also study Q-curvature equations in dimensions greater than 4 and obtain stronger results on the Hausdorff dimensions of the singular sets. More interestingly, our approach based on potential theory yields a significantly stronger finiteness theorem on the singular sets for Q-curvature equations in dimension 4, which is a remarkable analogue of Huber's theorem.
In this paper we present some extensions of the celebrated finite point conformal compactification theorem of Huber \cite{Hu57} for complete open surfaces to general dimensions based on the n-Laplace equations in conformal geometry. We are able to conclude a domain in the round sphere has to be the sphere deleted finitely many points if it can be endowed with a complete conformal metric with the negative part of the smallest Ricci curvature satisfying some integrable conditions. Our proof is based on the strengthened version of the Arsove-Huber's type theorem on n-superharmonic functions in our earlier work \cite{MQ18}. Moreover, using p-parabolicity, we push the injectivity theorem of Schoen-Yau to allow some negative curvature and therefore establish the finite point conformal compactification theorem for manifolds that have a conformal immersion into the round sphere. As a side product we establish the injectivity of conformal immersions from n-parabolicity alone, which is interesting by itself in conformal geometry.
We show the rigidity of the hexagonal Delaunay triangulated plane under Luo's PL conformality. As a consequence, we obtain a rigidity theorem for a particular type of locally finite convex ideal hyperbolic polyhedra.
In this paper we study asymptotic behavior of $n$-superharmonic functions at isolated singularity using the Wolff potential and $n$-capacity estimates in nonlinear potential theory. Our results are inspired by and extend those of Arsove-Huber and Taliaferro in 2 dimensions. To study $n$-superharmonic functions we use a new notion of $n$-thinness by $n$-capacity motivated by a type of Wiener criterion in Arsove-Huber's paper. To extend Taliaferro's work, we employ the Adams-Moser-Trudinger inequality for the Wolff potential, which is inspired by the one used by Brezis-Merle. For geometric applications, we study the asymptotic end behavior of complete conformally flat manifolds as well as complete properly embedded hypersurfaces in hyperbolic space. In both geometric applications the strong $n$-capacity lower bound estimate of Gehring in 1961 is brilliantly used. These geometric applications seem to elevate the importance of $n$-Laplace equations and make a closer tie to the classic analysis developed in conformal geometry in general dimensions.
In this paper, we construct Delaunay type constant mean curvature surfaces along a nondegenerate closed geodesic in a 3-dimensional Riemannian manifold.
Based on properties of n-subharmonic functions we show that a complete, noncompact, properly embedded hypersurface with nonnegative Ricci curvature in hyperbolic space has an asymptotic boundary at infinity of at most two points. Moreover, the presence of two points in the asymptotic boundary is a rigidity condition that forces the hypersurface to be an equidistant hypersurface about a geodesic line in hyperbolic space. This gives an affirmative answer to the question raised by Alexander and Currier in 1990.
In this paper we prove the conjecture of Alexander and Currier that states, except for covering maps of equidistant surfaces in hyperbolic 3-space, a complete, nonnegatively curved immersed hypersurface in hyperbolic space is necessarily properly embedded.
In this paper, we introduce a non linear ODE method to construct CMC surfaces in Riemannian manifolds with symmetry. As an application we construct unstable CMC spheres and outlying CMC spheres in asymptotically Schwarzschild manifolds with metrics like $g_{ij}=(1+\frac{1}{l})^{2}δ_{ij}+O(l^{-2})$. The existence of unstable CMC spheres tells us that the stability condition in Qing-Tian's work [Qing-Tian-CMC] can not be removed generally.
In this paper we consider the uniqueness problem of the constant mean curvature spheres in asymptotically flat 3-manifolds. We require the metric have the form g_{ij}=δ_{ij}+h_{ij} with h_{ij}=O_{4}(r^{-1}) and R=O(r^{-3-τ}),τ>0. We do not require the metric to be close to Schwarzschild metric in any sense or to satisfy RT conditions. We prove that, when the mass is not 0, stable CMC spheres that separate a certain compact part from infinity satisfy the radius pinching estimate r_{1}\leq Cr_{0} , which in many cases is critical to prove the uniqueness of the CMC spheres. As applications of this estimate, we remove the radius conditions of the uniqueness result in [Huang-CMC] and [NERZ-CMC] in some special cases.
In this paper I study the constant mean curvature surface in asymptotically flat 3-manifolds with general asymptotics. Under some weak condition, I prove that outside some compact set in the asymptotically flat 3-manifold with positive mass, the foliation of stable spheres of constant mean curvature is unique.