arXiv research

We produce longtime solutions to the Kähler-Ricci flow for complete Kähler metrics on $\Bbb C ^n$ without assuming the initial metric has bounded curvature, thus extending results in [3]. We prove the existence of a longtime bounded curvature solution emerging from any complete $U(n)$-invariant Kähler metric with non-n…

In this work, we obtain a existence criteria for the longtime Kähler Ricci flow solution. Using the existence result, we generalize a result by Wu-Yau on the existence of Kähler Einstein metric to the case with possibly unbounded curvature. Moreover, the Kähler Einstein metric with negative scalar curvture must be uniq…

Study on curvature flow in Minkowski space for cocompact hypersurfaces.

New flow solves LYZ equation on Kähler manifolds.

New method finds metrics on surfaces with prescribed curvatures using circle packings and surgery.

Study proves mean curvature flow in GRW spacetimes with perpendicular boundary condition.

We study convex entire graphs evolving with normal velocity equal to a positive power of the mean curvature. Under mild assumptions we prove longtime existence.

We establish the longtime existence and convergence results of the mean curvature flow of entire Lagrangian graphs in Pseudo-Euclidean space which is related to Logarithmic gradient flow.

We consider a fully nonlinear parabolic equation with nonlinear Neumann type boundary condition, and show that the longtime existence and convergence of the flow. Finally we apply this study to the boundary value problem for minimal Lagrangian graphs.

Fractional combinatorial flow improves surface conformal structures.

The paper establishes a discrete uniformization theorem for surfaces with piecewise hyperbolic metrics.

In this paper we study the Ricci flow on surfaces homeomorphic to a cylinder (that is, a product of the circle with a compact interval). We prove longtime existence results, results on the asymptotic behavior of the flow, and we report on an interesting phenomenon: convergence to constant curvature in the normalised fl…

We prove the longtime existence and convergence of the Calabi flow on toric Fano surfaces in a large family of Kahler classes where the class has positive extremal Hamiltonian potential and the initial Calabi energy is bounded by some constant. This is an extension of our previous work. We use the toric condition in a …

We prove longtime existence and estimates for solutions to a fully nonlinear Lagrangian parabolic equation with locally $C^{1,1}$ initial data $u_0$ satisfying either (1) $-(1+η) I_n\leq D^2u_0 \leq (1+η)I_n$ for some positive dimensional constant $η$, (2) $u_0$ is weakly convex everywhere or (3) $u_0$ satisfies a larg…

Smooth solutions found for modified mean curvature flow in Riemannian manifolds.

In this paper, we study a mean curvature type flow with capillary boundary in the unit ball. Our flow preserves the volume of the bounded domain enclosed by the hypersurface, and monotonically decreases an energy functional $E$. We show that it has the longtime existence and subconverges to spherical caps. As an applic…

We consider inverse curvature flows in warped product manifolds, which are constrained subject to local terms of lower order, namely the radial coordinate and the generalized support function. Under various assumptions we prove longtime existence and smooth convergence to a coordinate slice. We apply this result to ded…

In this article, we introduce a new type of mean curvature flow for bounded star-shaped domains in space forms and prove its longtime existence, exponential convergence without any curvature assumption. Along this flow, the enclosed volume is a constant and the surface area evolves monotonically. Moreover, for a bounde…

The study proves stability of various graphical translators in mean curvature flow.

Proves existence of circle patterns on surfaces with cusps.

The paper proves geometric inequalities in sphere using locally constrained flows.

In this paper we study the Type IIb mean curvature flow. We first prove that if the convex entire graph $(y,u(|y|))$ over $\mathbb{R}^n$, $n\geq 2$, satisfying there exist positive constants $ε$, $c$ and $N$ such that $u'(r)\geq c r^ε$ for $r\geq N$, the longtime solution to mean curvature flow with initial data $(y,…

In [8] Gerhardt proves longtime existence for the inverse mean curvature flow in globally hyperbolic Lorentzian manifolds with compact Cauchy hypersurface, which satisfy three main structural assumptions: a strong volume decay condition, a mean curvature barrier condition and the timelike convergence condition. Further…

The purpose of this paper is to give a self-contained exposition of the Atiyah-Bott picture for the Yang-Mills equation over Riemann surfaces with an emphasis on the analogy to finite dimensional geometric invariant theory. The main motivation is to provide a careful study of the semistable and unstable orbits: This in…

We consider mean curvature flow of an initial surface that is the graph of a function over some domain of definition in $R^n$. If the graph is not complete then we impose a constant Dirichlet boundary condition at the boundary of the surface. We establish longtime-existence of the flow and investigate the projection of…

The paper studies curvature flows in Euclidean and hyperbolic spaces, proving smooth convergence to spheres.

Study on deforming discrete conformal structures on surfaces with boundaries.

The paper finds hyperbolic metrics on surfaces with boundary using combinatorial curvature flows.

We reduce the embedding problem for hypo SU(2) and SU(3)-structures to the embedding problem for hypo G2-structures into parallel Spin(7)-manifolds. The latter will be described in terms of gauge deformations. This description involves the intrinsic torsion of the initial G2-structure and allows us to prove that the ev…

The paper studies rigid sphere packings on 3D manifolds with boundary.

We prove a monotonicity formula for mean curvature flow with surgery. This formula differs from Huisken's monotonicity formula by an extra term involving the mean curvature. As a consequence, we show that a surgically modified flow which is sufficiently close to a smooth flow in the sense of geometric measure theory is…

First example of open manifold with positive Ricci curvature and non-proper Busemann function.

Paper introduces new flows to find circle packings with specific curvature.

The paper develops algorithms for finding metrics with prescribed combinatorial curvature on polyhedral surfaces.

We consider the mean curvature flow of entire Lagrangian graphs with Lipschitz continuous initial data. Assuming only a certain bound on the Lipschitz norm of an initial entire Lagrangian graph in $\R^{2n}$, we show that the parabolic equation \eqref{PMA} for the Lagrangian potential has a longtime solution which is sm…

The paper proves Michael-Simon inequalities in hyperbolic space using novel curvature flows.

Paper studies inverse curvature flows and solves related geometric problems.

Study on existence and structure of P-area surfaces in Heisenberg group.

Study finds both existence and non-existence of maps in Morse boundaries.

Study on Lane-Emden equation on curved spaces, revealing new existence and non-existence phenomena.

The $(α,β)$-Ricci-Yamabe flow exists on closed manifolds.

Proves generic existence of spectral networks for many cases.

In this article, we shall investigate the relationship between the existence or non-existence of non-singular solutions to the normalized Ricci flow and smooth structures on closed 4-manifolds, where non-singular solutions to the normalized Ricci flow are solutions which exist for all time $t \in [0, \infty)$ with unif…

The paper studies equations on almost Hermitian manifolds with estimates and existence results.

Study abelianization of Lie algebroids and groupoids, providing conditions for existence.

The paper examines conditions for Einstein multiply warped products and estimates their parameters.

New methods prove existence of rotating shapes moving in space.

A simplified proof of the Alexander-Conway polynomial exists.