arXiv research

New Einstein manifolds split into symmetric and compact parts.

Under an infinitesimal version of the Bishop-Gromov relative volume comparison condition for a measure on an Alexandrov space, we prove a topological splitting theorem of Cheeger-Gromoll type. As a corollary, we prove an isometric splitting theorem for Riemannian manifolds with singularities of nonnegative (Bakry-Emery…

The paper proves integral formulas for manifolds with multiple orthogonal distributions.

A maximum principle for C^0 null hypersurfaces is obtained and used to derive a splitting theorem for spacetimes which contain null lines. As a consequence of this null splitting theorem, it is proved that an asymptotically simple vacuum (Ricci flat) spacetime which contains a null line is isometric to Minkowski space.

The paper examines perimeter minimizing sets in curved spaces and finds conditions for their boundary to match a specific structure.

In this paper, I prove a splitting theorem for equifocal submanifolds with non-flat section in a simply connected symmetric space of compact type. Also, by using the splitting theorem, I prove that the sections of equifocal submanifolds with non-flat section in an irreducible simply connected symmetric space of compact…

New definition of Bäcklund transformation for surface isometric deformation.

We investigate the structure of a Finsler manifold of nonnegative weighted Ricci curvature including a straight line, and extend the classical Cheeger-Gromoll-Lichnerowicz splitting theorem. Such a space admits a diffeomorphic, measure-preserving splitting in general. As for a special class of Berwald spaces, we can pe…

We characterize those spacetimes which admit a isometric (or conformal) embedding in some Lorentz-Minkowski space L^N. In particular, any globally hyperbolic spacetime can be isometrically embedded in L^N. This is proven by a result of its own interest: the construction of a smooth time function whose gradient is bound…

The paper proves conditions for Einstein solitons to split into line and manifold.

Sharp spectral theorem splits certain non-compact manifolds.

Study rigidity of spectral gap on Finsler manifolds with specific curvature bounds.

The paper examines rigidity of metric constructions in Wasserstein spaces.

New condition prevents hyperbolic spaces from matching curve complexes.

New theorem splits weighted Lorentz-Finsler manifolds into simpler parts.

The paper proves a theorem about splitting manifolds with specific curvature properties.

By using a Borel density theorem for algebraic quotients, we prove a theorem concerning isometric actions of a Lie group $G$ on a smooth or analytic manifold $M$ with a rigid $\mathrm{A}$-structure $σ$. It generalizes Gromov's centralizer and representation theorems to the case where $R(G)$ is split solvable and $G/R(G…

We show that noncompact simply connected harmonic manifolds with volume density $Θ_{p}(r) =\sinh ^{n-1} r$ is isometric to the real hyperbolic space and noncompact simply connected Kähler harmonic manifold with volume density $Θ_{p}(r) =\sinh ^{2n-1} r \cosh r$ is isometric to the complex hyperbolic space. A similar re…

Researchers prove a 30-year-old cosmological conjecture about spacetime.

Cao's splitting theorem says that for any complete Kähler-Ricci flow $(M,g(t))$ with $t\in [0,T)$, $M$ simply connected and nonnegative bounded holomorphic bisectional curvature, $(M,g(t))$ is holomorphically isometric to $\C^k\times (N,h(t))$ where $(N,h(t))$ is a Kahler-Ricci flow with positive Ricci curvature for $t…

Let $X=G/K$ be a higher rank symmetric space of non-compact type, where $G$ is the connected component of the isometry group of $X$. We define the splitting rank of $X$, denoted by $\text{srk}(X)$, to be the maximal dimension of a totally geodesic submanifold $Y\subset X$ which splits off an isometric $\mathbb R$-facto…

We consider isometric immersions into space forms having the second fundamental form parallel at order k. We show that this class of immersions consists of local products, in a suitably defined sense, of parallel immersions and normally flat immersions of flat spaces.

Affirmative answer to splitting question for open manifolds with nonnegative Ricci curvature and linear volume growth.

The paper extends a splitting theorem for a specific type of tensor in Riemannian geometry.

We prove a result on equivariant deformations of flat bundles, and as a corollary, we obtain two ``splitting in a finite cover'' theorems for isometric group actions on Riemannian manifolds with infinite fundamental groups, where the manifolds are either compact of nonnegative Ricci curvature, or complete of nonnegativ…

Given a finite graph of relatively hyperbolic groups with its fundamental group relatively hyperbolic and edge groups quasi-isometrically embedded and relatively quasiconvex in vertex groups, we prove that vertex groups are relatively quasiconvex if and only if all the vertex groups have finite relative height in the f…

We prove that if a geodesic metric measure space satisfies a comparison condition for isoperimetric profile and if the observable variance is maximal, then the space is foliated by minimal geodesics, where the observable variance is defined to be the supremum of the variance of 1-Lipschitz functions on the space. Our r…

We characterize complete nonnegatively curved steady gradient soliton with curvature in L^1. We show that there are isometric to a product (R^2,g_{cigar}) times(R^{n-2}, eucl))/Gamma where Gamma is a Bieberbach group of rank n-2. We prove also a similar local splitting result under weaker curvature assumptions.

Metric spaces with certain curvature properties are universally infinitesimally Hilbertian.

The energy of any $C^1$ representative of a homotopy class of maps from a compact and connected Riemannian manifold with nonnegative Ricci curvature into a complete Riemannian manifold with no conjugate points is bounded below by a constant determined by the asymptotic geometry of the target, with equality if and only …

Decomposes Q-Fano Kähler-Einstein varieties into simpler components.

The paper proves manifold splitting theorems with nonnegative intermediate curvature.

In this paper we prove that on a complete smooth metric measure space with non-negative Bakry-Émery-Ricci curvature if the space of weighted L^2 harmonic one-forms is non-trivial then the weighted volume of the manifold is finite and universal cover of the manifold splits isometrically as the product of the real line w…

We show that any group that is hyperbolic relative to virtually nilpotent subgroups, and does not admit peripheral splittings, contains a quasi-isometrically embedded copy of the hyperbolic plane. In natural situations, the specific embeddings we find remain quasi-isometric embeddings when composed with the inclusion m…

We prove that any 4-dimensional geodesically complete spacetime with a timelike Killing field satisfying the vacuum Einstein field equation $Ric(g_{M})=λg_{M}$ with nonnegative cosmological constant $λ\geq 0$ is flat. When dim $\geq 5$, if the spacetime is assumed to be static additionally, we prove that its universal …

The study embeds infinite-dimensional geometric structures in Cayley graphs.

Cohomogeneity-one actions on symmetric spaces of mixed type

The pants graph of a free group is constructed and studied.

Unified representation for tree ensembles indexed by nodes

We show if M is a closed, connected, orientable, hyperbolic 3-manifold with Heegaard genus g then g >= 1/2 cosh(r) where r denotes the radius of any isometrically embedded ball in M. Assuming an unpublished result of Pitts and Rubinstein improves this to g >= 1/2 cosh(r) + 1/2. We also give an upper bound on the volume…

We derive some consequences of the Liouville theorem for plurisubharmonic functions of L.-F. Tam and the author. The first result provides a nonlinear version of the complex splitting theorem (which splits off a factor of $\mathbb{C}$ isometrically from the simply-connected Kähler manifold with nonnegative bisectional …

The study examines gradient Ricci solitons with nonnegative curvature, proving properties of their blow-downs.

The paper describes hyperkähler geometry of cotangent bundles using rank-1 projections.

The main purpose of these lecture notes is to provide a concise introduction to Lie groups, Lie algebras, and isometric and adjoint actions, aiming mostly at advanced undergraduate and graduate students. In addition, the connection between such classic theories and the research area of the first author is explored. Nam…

We study locally conformally Berwald metrics on closed manifolds which are not globally conformally Berwald. We prove that the characterization of such metrics is equivalent to characterizing incomplete, simply-connected, Riemannian manifolds with reducible holonomy group whose quotient by a group of homotheties is clo…

Let $(M^n, g)$ be a compact Kähler manifold with nonpositive bisectional curvature. We show that a finite cover is biholomorphic and isometric to a flat torus bundle over a compact Kähler manifold $N^k$ with $c_1 < 0$. This confirms a conjecture of Yau. As a corollary, for any compact Kähler manifold with nonpositive b…

In \cite{LiWang2001complete1,LiWang2001complete2}, Li-Wang proved a splitting theorem for an n-dimensional Riemannian manifold with $Ric\geqslant -(n-1)$ and the bottom of spectrum $λ_0(M)=\frac{(n-1)^2}{4}$. For an n-dimensional compact manifold $M$ with $Ric\geqslant-(n-1)$ with the volume entropy $h(M)=n-1$, Ledrapp…

In this paper we consider the Ricci curvature of a Ricci soliton. In particular, we have showed that a complete gradient Ricci soliton with non-negative Ricci curvature possessing a non-constant convex potential function having finite weighted Dirichlet integral satisfying an integral condition is Ricci flat and also i…