arXiv subjects
Publications and source records attributed to Mijia Lai.
In this paper, based on the warped product structures determined by the Obata-type equations with Robin boundary condition, we establish some rigidity results for compact manifolds with smooth boundary under appropriate Bakry-Émery Ricci curvature conditions and some other assumptions.
Let $\mathcal{H}_d(M)$ denote the space of harmonic functions with polynomial growth of degree at most $d$ on a complete Riemannian manifold $(M,g)$. Yau raised two fundamental questions regarding $\mathcal{H}_d(M)$ on complete manifolds with nonnegative Ricci curvature. The first question is the finite dimensionality of $\mathcal{H}_d(M)$, which was confirmed by Colding and Minicozzi. The second question asks whether a sharp upper bound given by its Euclidean analog $\operatorname{dim}\mathcal{H}_{d}(\mathbb{R}^n)$ holds. We verify that the second question is true on locally conformally flat manifolds. Indeed, one can precisely determine the value of $\dim \mathcal{H}_d(M)$ case by case.
In this paper, we establish a necessary condition for the logarithmic Minkowski problem in higher dimensions. This result generalizes a necessary condition proposed by Liu, Lu, Sun, and Xiong in their investigation of the two-dimensional case, and also refines the so-called subspace concentration condition.
We verify a conjecture proposed by X. Chen and Y. Shi, which arises from their study of the Green function on spheres in Euclidean space. More precisely, let $M\subset \mathbb{R}^3$ be a closed $C^{2}$ embedded surface and suppose that there exists a point $p\in M$ so that its Green function $G$ is of the form $G(p,q)=-\frac{1}{2π} \ln d_{\mathbb{R}^3}(p,q)+c, \forall q\neq p$, then $M$ must be a round sphere.
We establish an area growth estimate for solutions that are bounded from above of the Liouville equation $Δu+K e^{2u}=0$ with a positive pinched curvature $0<λ\leq K\leqΛ$. As an application, we provide a new proof of Eremenko-Gui-Li-Xu's result in [EGLX]. We also classify solutions with an upper bound in the half plane with the boundary having constant geodesic curvature.
We study finite total curvature solutions of the Liouville equation $Δu+e^{2u}=0$ on a complete surface $(M,g)$ with nonnegative Gauss curvature. It turns out that the asymptotic behavior of the solution separates two extremal cases: on the one end, if the solution decays not too fast, then $(M,g)$ must be isometric to the standard Euclidean plane; on the other end, if $(M,g)$ is isometric to the flat cylinder $\mathbb{S}^1\times \mathbb{R}$, then solutions must decay linearly and are completely classified.
We obtain some rigidity results for metrics whose Schouten tensor is bounded from below after conformal transformations. Liang Cheng recently proved that a complete, nonflat, locally conformally flat manifold with Ricci pinching condition ($Ric-εRg\geq 0$) must be compact. This answers higher dimensional Hamilton's pinching conjecture on locally conformally flat manifolds affirmatively. Since (modified) Schouten tensor being nonnegative is equivalent to a Ricci pinching condition, our main result yields a simple proof of Cheng's theorem.
We give an interpretation of the hemisphere rigidity theorem of Hang-Wang in the framework of Gelfand problem. More precisely, Hang-Wang showed that for a metric $g$ conformal to the standard metric $g_0$ on $S^{n}_{+}$ with $R\geq n(n-1)$ and whose boundary coincides with $g_0|_{\partial S^{n}_{+}}$, then $g=g_0$. This is related to the classical Gelfand problem, which investigates $-Δu=λg(u)$ for certain nonlinearity $g$ in a bounded region $Ω\subset \mathbb{R}^n$ subject to the Dirichlet boundary condition. It is well-known that there exists an extremal $λ^{*}$, such that for $λ>λ^{*}$, the above equation does not admit any solution. Interestingly, Hang-Wang's hemisphere rigidity theorem yields a precise value for $λ^{*}$ for $g(u)=e^{2u}$ when $n=2$ and $g(u)=(1+u)^{\frac{n+2}{n-2}}$ for $n\geq 3$. We attempt to generalize the hemisphere rigidity theorem under $Q$ curvature lower bound and fit this into the interpretation of fourth order Gelfand problem for bi-Laplacian with conformal nonlinearity.
Let $(M,g)$ be a compact manifold with boundary and $Ric_g\geq (n-1)g$, Hang and Wang proved that $(M,g)$ is isometric to the standard hemisphere if $\partial M$ is convex and isometric to $\mathbb{S}^{n-1}(1)$. We prove some rigidity theorems when $\partial M $ is isometric to a product manifold where one factor is the standard sphere.
We study the Obata equation with Robin boundary condition $\frac{\partial f}{\partial ν}+af=0$ on manifolds with boundary, where $a \in \mathbb{R}\setminus\{0\}$. Dirichlet and Neumann boundary conditions were previously studied by Reilly \cite{R}, Escobar \cite{Es} and Xia \cite{X}. Compared with their results, the sign of $a$ plays an important role here. The new discovery shows besides spherical domains, there are other manifolds for both $a>0$ and $a<0$. We also consider the Obata equation with non-vanishing Neumann condition $\frac{\partial f}{\partial ν}=1$.
Let $(X^{n},g_+) $ $(n\geq 3)$ be a Poincaré-Einstein manifold which is $C^{3,α}$ conformally compact with conformal infinity $(\partial X, [\hat{g}])$. On the conformal compactification $(\overline{X}, \bar g=ρ^2g_+)$ via some boundary defining function $ρ$, there are two types of Yamabe constants: $Y(\overline{X},\partial X,[\bar g])$ and $Q(\overline{X},\partial X,[\bar g])$. (See definitions (\ref{def.type1}) and (\ref{def.type2})). In \cite{GH}, Gursky and Han gave an inequality between $Y(\overline{X},\partial X,[\bar g])$ and $Y(\partial X,[\hat{g}])$. In this paper, we first show that the equality holds in Gursky-Han's theorem if and only if $(X^{n},g_+)$ is isometric to the standard hyperbolic space $(\mathbb{H}^{n}, g_{\mathbb{H}})$. Secondly, we derive an inequality between $Q(\overline{X},\partial X,[\bar g])$ and $Y(\partial X, [\hat g])$, and show that the equality holds if and only if $(X^{n},g_+)$ is isometric to $(\mathbb{H}^{n}, g_{\mathbb{H}})$. Based on this, we give a simple proof of the rigidity theorem for Poincaré-Einstein manifolds with conformal infinity being conformally equivalent to the standard sphere.
We obtain sharp volume bound for a conic 2-sphere in terms of its Gaussian curvature bound. We also give the geometric models realizing the extremal volume. In particular, when the curvature is bounded in absolute value by $1$, we compute the minimal volume of a conic sphere in the sense of Gromov. In order to apply the level set analysis and iso-perimetric inequality as in our previous works, we develop some new analytical tools to treat regions with vanishing curvature.
We study metrics on conic 2-spheres when no Einstein metrics exist. In particular, when the curvature of a conic metric is positive, we obtain the best curvature pinching constant. We also show that when this best pinching constant is approached, the conic 2-sphere has an explicit Gromov-Hausdorff limit. This is a generalization of the previous results of Chen-Lin and Bartolucci for 2-spheres with one or two conic points.
We show that spheres of positive constant curvature with $n$ ($n\geq3$) conic points converge to a sphere of positive constant curvature with two conic points (or called an (American) football) in Gromov-Hausdorff topology when the corresponding singular divisors converge to a critical divisor in the sense of Troyanov. We prove this convergence in two different ways. Geometrically, the convergence follows from Luo-Tian's explicit description of conic spheres as boundaries of convex polytopes in $S^{3}$. Analytically, regarding the conformal factors as the singular solutions to the corresponding PDE, we derive the required a priori estimates and convergence result after proper reparametrization.
We study the J-flow on Kahler surfaces when the Kahler class lies on the boundary of the open cone for which global smooth convergence holds, and satisfies a nonnegativity condition. We obtain a C^0 estimate and show that the J-flow converges smoothly to a singular Kahler metric away from a finite number of curves of negative self-intersection on the surface. We discuss an application to the Mabuchi energy functional on Kahler surfaces with ample canonical bundle.
We study the convergence behavior of the general inverse $σ_k$-flow on Kähler manifolds with initial metrics satisfying the Calabi Ansatz. The limiting metrics can be either smooth or singular. In the latter case, interesting conic singularities along negatively self-intersected sub-varieties are formed as a result of partial blow-up.
In this note, we extend our previous work on the inverse $σ_k$ problem. Inverse $σ_{k}$ problem is a fully nonlinear geometric PDE on compact Kähler manifolds. Given a proper geometric condition, we prove that a large family of nonlinear geometric flows converges to the desired solution of the given PDE.
In this paper, we study a class of fully nonlinear metric flow on Kähler manifolds, which includes the J-flow as a special case. We provide a sufficient and necessary condition for the long time convergence of the flow, generalizing the result of Song-Weinkove. As a consequence, under the given condition, we solved the corresponding Euler equation, which is fully nonlinear of Monge-Ampère type. As an application, we also discuss a complex Monge-Ampère type equation including terms of mixed degrees, which was first posed by Chen.