arXiv research

Study complete manifolds with weighted Poincaré inequality and Ricci curvature bounds.

New proof given for a functional's minimum condition.

We prove that a quasiisometric map between rank one symmetric spaces is within bounded distance from a unique harmonic map. In particular, this completes the proof of the Schoen-Li-Wang conjecture.

The paper proves criticality criteria and spectral splitting theorems for manifolds with Ricci bounds.

Study on exact Lagrangian submanifolds in unit ball with Legendrian boundary.

We study manifolds satisfying a weighed Poincare inequality, which was first introduced by Li-Wang. We generalized one of their results by relaxing the Ricci curvature bound condition only being satisfied outside a compact set and established a finitely many ends result. We proved a vanishing result for $L^2$ harmonic …

This paper contains some vanishing theorems for $L^2$ harmonic forms on complete Riemannian manifolds with a weighted Poincaré inequality and a certain lower bound of the curvature. The results are in the spirit of Li-Wang and Lam, but without assumptions of sign and growth rate of the weight function, so they can be a…

We prove that the isoperimetric inequality is satisfied in the cigar steady soliton and in the Bryant steady soliton. Since both of them are Riemannian manifolds with warped product metric, we utilize the result of Guan-Li-Wang to get our conclusion. For the sake of the soliton structure, we believe that the geometric …

We first proved a compactness theorem of the Kähler metrics, which confirms a prediction of Chen. Then we prove several eigenvalue estimates along the Calabi flow. Combining the compactness theorem and these eigenvalue estimates, we generalize the method developed by Chen-Li-Wang to prove the small energy theorems of t…

We study dynamic optimal portfolio allocation for monotone mean--variance preferences in a general semimartingale model. Armed with new results in this area we revisit the work of Cui, Li, Wang and Zhu (2012, MAFI) and fully characterize the circumstances under which one can set aside a non-negative cash flow while sim…

New proof of splitting theorem and finite ends of minimal hypersurfaces in nonnegative curvature manifolds.

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 immersed two-sided minimal hypersurfaces in $\mathbb{R}^n$ with finite total curvature. We prove that the sum of the Morse index and the nullity of the Jacobi operator is bounded from below by a linear function of the number of ends and the first Betti number of the hypersurface. When $n=4$, …

In this paper, we study vanishing and splitting results on a complete smooth metric measure space $(M^n,g,\mathrm{e}^{-f}\mathrm{d}v)$ with various negative $m$-Bakry-Émery-Ricci curvature lower bounds in terms of the first spectrum $λ_1(Δ_f)$ of the weighted Laplacian $Δ_f$, i.e. $\mathrm{Ric}_{m,n}\geq -aλ_1(Δ_f)-b$ …

In this paper we develop new methods for studying the convergence problem for the heat flow on negatively curved spaces and prove that any quasiconformal map of the sphere $\mathbb{S}^{n-1}$, $n\geq 3$, can be extended to the $n$-dimensional hyperbolic space such that the heat flow starting with this extension converge…

We construct a canonical Hausdorff complex analytic moduli space of Fano manifolds with Kähler-Ricci solitons. This naturally enlarges the moduli space of Fano manifolds with Kähler-Einstein metrics, which was constructed by Odaka and Li-Wang-Xu. We discover a moment map picture for Kähler-Ricci solitons, and give comp…

Let $(Σ,g)$ be a closed Riemannian surface, $\textbf{G}=\{σ_1,\cdots,σ_N\}$ be an isometric group acting on it. Denote a positive integer $\ell=\inf_{x\inΣ}I(x)$, where $I(x)$ is the number of all distinct points of the set $\{σ_1(x),\cdots,σ_N(x)\}$. A sufficient condition for existence of solutions to the mean field …

Study confirms conjecture: minimal Lagrangian surfaces with Legendrian boundary are rigid.

Let $(M,g)$ be a compact Riemannian surface without boundary, $W^{1,2}(M)$ be the usual Sobolev space, $J: W^{1,2}(M)\rightarrow \mathbb{R}$ be the functional defined by $J(u)=\frac{1}{2}\int_M|\nabla u|^2dv_g+8π\int_M udv_g-8π\log\int_Mhe^udv_g,$ where $h$ is a positive smooth function on $M$. In an inspiring work (…

Study geometric structure of Ricci shrinker ends without global curvature assumptions.

The study classifies and proves rigidity of Legendrian self-shrinkers in 3D and 5D.

The paper extends isometric embedding results to null cones and spheres.

Solves a complex Monge-Ampère equation on compact Hermitian manifolds.

Let $(M^n, g, e^{-f}dv)$ be a smooth metric measure space of dimensional $n$. Suppose that $v$ is a positive weighted $p$-eigenfunctions associated to the eigenvalues $λ_{1,p}$ on $M$, namely $e^{f}div(e^{-f}|\nabla v|^{p-2}\nabla v)=-λ_{1,p}v^{p-1}.$ in the distribution sense. We first give a local gradient estimat…

The paper develops techniques to study entropy and rigidity in RCD-spaces.

Paper extends isoperimetric inequalities for non-starshaped hypersurfaces.

New equations reveal viscosity from boundary measurements.

The paper defines a universal Teichmüller space for PGL_d(R) and proves its properties.

Let $M$ be an $n(>2)$-dimensional closed orientable submanifold in an $(n+p)$-dimensional space form $\mathbb{R}^{n+p}(c)$. We obtain an optimal upper bound for the second eigenvalue of a class of elliptic operators on $M$ defined by $L_{T}f=-div(T\nabla f)$, where $T$ is a general symmetric, positive definite and dive…

The paper solves a mean field equation on a compact Riemann surface using variational and blowup analysis.

The paper proves existence of solutions for mean field equations on compact Riemann surfaces.

The paper proves rigidity results for compact initial data sets.

New Liouville-type results for CR Yamabe equation in Heisenberg group.

In this note we give geometric formulations and proofs of three results of S. Morita. These results relate certain two dimensional cohomology classes of various moduli spaces of curves. We also give a geometric interpretation of a fourth result of Morita. One motivation of this work is to facilitate the application of …

Study proves structure results for homogeneous spaces supporting specific equations.

We prove a rigidity result in the sphere which allows us to generalize a result about smooth convex hypersurfaces in the sphere by Do Carmo-Warner to convex $C^2$-hypersurfaces. We apply these results to prove $C^{1,β}$-convergence of inverse F-curvature flows in the sphere to an equator in \mathbb{S}^{n+1} for embedde…

New rigidity results for specific hypersurfaces in spacetimes.

This article studies the nonabelian localization results of Beasley and Witten, and considers the analogue of these results when the gauge group is U(1). It compares these results with results of Manoliu on abelian Chern-Simons theory, showing that the dependence on the coupling constant is the same.

We obtain two types of results on positive scalar curvature metrics for compact spin manifolds that are even dimensional. The first type of result are obstructions to the existence of positive scalar curvature metrics on such manifolds, expressed in terms of end-periodic eta invariants that were defined by Mrowka-Ruber…

We extend recent results of Guan and Spruck, proving existence results for constant Gaussian curvature hypersurfaces in Hadamard manifolds.

New results on splitting tangles and spatial graphs.

The paper surveys mathematical results on filtration enlargement with financial examples.

The paper studies rigidity results for harmonic forms on Kähler manifolds.

Paper proves symphonic map result similar to Eells-Sampson.

LLE produces unwanted results without regularization, which can be prevented with regularization.

Paradan and Vergne generalised the quantisation commutes with reduction principle of Guillemin and Sternberg from symplectic to Spin$^c$-manifolds. We extend their result to noncompact groups and manifolds. This leads to a result for cocompact actions, and a result for non-cocompact actions for reduction at zero. The r…

In this paper we show a quantitative rigidity result for the minimizer of the Willmore functional among all projective planes in $\mathbb{R}^n$ with $n\ge 4$. We also construct an explicit counterexample to a corresponding rigidity result in codimension one, by showing that an Enneper surface might split-off during a b…

New stability and isolation results for Einstein manifolds.