arXiv research

Develops local curvature estimates for mean curvature flow.

Global and local estimates for a curvature equation on manifolds with boundary.

Proves local noncollapsing estimate for mean curvature flow.

Localizes curvature estimates for evolving hypersurfaces under various flows.

Study local curvature estimates and existence of conformal metrics on noncompact manifolds.

Paper proves curvature estimates for a specific flow on Kähler manifolds.

Estimates Kähler metrics with noncollapsing volume under complex Monge-Ampère constraints.

Assuming local uniform bounds on the metric for a solution of the Chern-Ricci flow, we establish local Calabi and curvature estimates using the maximum principle.

We study curvature pinching estimates of Ricci flow on complete 3- dimensional manifolds without bounded curvature assumption. We will derive some general curvature conditions which are preserved on any complete solution of 3-dim Ricci flow, these conditions include nonnegative Ricci curvature and sectional curvature a…

Curvature estimate for stable free boundary minimal hypersurfaces in wedge-shaped manifolds.

The paper estimates curvature for a specific flow on manifolds.

Study local curvature of Kähler-Ricci flow on semi-ample manifolds.

We obtain a local Sobolev constant estimate for integral Ricci curvature, which enables us to extend several important tools such as the maximal principle, the gradient estimate, the heat kernel estimate and the $L^2$ Hessian estimate to manifolds with integral Ricci lower bounds, without the non-collapsing conditions.

Paper bounds local curvature for Ricci-harmonic flow under Ricci bounded condition.

Estimates spacelike surfaces' curvature in de Sitter space.

Estimates mean curvature, scalar curvature, shape operator in warped products.

Estimates manifold dimension using local graph structure.

Estimates mean curvature flow with geometric bounds.

In this paper we present several curvature estimates for solutions of the Ricci flow which depend on smallness of certain local integrals of the norm of the Riemann curvature tensor.

In this paper, we prove the local gradient estimate for harmonic functions on complete, noncompact Finsler measure spaces under the condition that the weighted Ricci curvature has a lower bound. As applications, we obtain Liouville type theorem on Finsler manifolds with nonnegative Ricci curvature.

In this paper we give local curvature estimates for the Laplacian flow on closed G_2-structures under the condition that the Ricci curvature is bounded along the flow. The main ingredient consists of the idea of Kotschwar-Munteanu-Wang who gave local curvature estimates for the Ricci flow on complete manifolds and then…

The paper extends Perelman's theorems on Ricci flow entropy.

Study shows how weak inverse anisotropic mean curvature flow behaves at infinity.

The study of the $k$-th elementary symmetric function of the Weyl-Schouten curvature tensor of a Riemannian metric, the so called $σ_k$ curvature, has produced many fruitful results in conformal geometry in recent years, especially when the dimension of the underlying manifold is 3 or 4. In these studies, the deforming…

Establishes refined singularity estimate for nonnegative n-superharmonic functions in locally conformally flat manifolds.

We give lower bounds for the fundamental tone of open sets in submanifolds with locally bounded mean curvature in $N \times \mathbb{R}$, where $N$ is an $n$-dimensional complete Riemannian manifold with radial sectional curvature $K_{N} \leq κ$. When the immersion is minimal our estimates are sharp. We also show that …

Study finds solutions to curvature equation with boundary conditions.

Ricci flow smooths locally collapsing manifolds with controlled curvature.

Note establishes a local maximum principle for Ricci flow under curvature conditions.

CA-PCA improves manifold dimension estimation by accounting for curvature.

In this paper we present several curvature estimates and convergence results for solutions of the Ricci flow. The curvature estimates depend on smallness of certain local space-time integrals of the norm of the Riemann curvature tensor, while the convergence results require finiteness of space-time integrals of the nor…

Study finds strictly convex surfaces with specific curvature and boundary in space forms.

We give lower bounds for the first Dirichilet eigenvalues for domains in submanifolds with locally bounded mean curvatures. These bounds depend on the injectivity radius, sectional curvature (upperbound) of the ambient space and on the mean curvature of the submanifold. For submanifolds fo Hadamard manifolds these lowe…

This paper is devoted to $C^2$ a priori estimates for strictly locally convex radial graphs with prescribed Weingarten curvature and boundary in space forms. By constructing two-step continuity process and applying degree theory arguments, existence results in space forms are established for prescribed Gauss curvature …

Optimizes heat equation estimates on noncompact manifolds.

By using the De Giorgi iteration method we will give a new simple proof of the recent result of B.Kotschwar, O.Munteanu, J.Wang [KMW] and N.Sesum [S] on the local boundedness of the Riemmanian curvature tensor of solutions of Ricci flow in terms of its inital value on a given ball and a local uniform bound on the Ricci…

We show that the norm of the Riemann curvature tensor of any smooth solution to the Ricci flow can be explicitly estimated in terms of its initial values on a given ball, a local uniform bound on the Ricci tensor, and the elapsed time. This provides a new, direct proof of a result of Sesum, which asserts that the curva…

Estimates Bartnik mass for metrics with nonnegative Gauss curvature.

Sharp bounds on diameter and eigenvalues for amply regular graphs.

In this note, we obtain a sharp volume estimate for complete gradient Ricci solitons with scalar curvature bounded below by a positive constant. Using Chen-Yokota's argument we obtain a local lower bound estimate of the scalar curvature for the Ricci flow on complete manifolds. Consequently, one has a sharp estimate of…

The paper proves gradient estimates for a weighted p-Laplacian equation on Riemannian manifolds.

We derive local $C^{2}$ estimates for complete non-compact translating solitons of the Gauss curvature flow in $\mathbb{R}^3$ which are graphs over a convex domain $Ω$. This is closely is related to deriving local $C^{1,1}$ estimates for the degenerate Monge-Ampére equation. As a result, given a weakly convex bounded d…

We prove certain localized and global differential Harnack inequality for all positive solutions to the geometric conjugate heat equation coupled to the forward in time Ricci flow. In this case, the diffusion operator is perturbed with the curvature operator, precisely, the Laplace-Beltrami operator is replaced with "$…

In this paper, we derive apriori estimates for constant scalar curvature Kähler metrics on a compact Kähler manifold. We show that higher order derivatives can be estimated in terms of a $C^0$ bound for the Kähler potential. We also discuss some local versions of these estimates which can be of independent interest.

In this paper we study the geometry of first time singularities of the mean curvature flow. By the curvature pinching estimate of Huisken and Sinestrari, we prove that a mean curvature flow of hypersurfaces in the Euclidean space $\R^{n+1}$ with positive mean curvature is $κ$-noncollapsing, and a blow-up sequence conve…

We review recent work on the local geometry and optimal regularity of Lorentzian manifolds with bounded curvature. Our main results provide an estimate of the injectivity radius of an observer, and a local canonical foliations by CMC (Constant Mean Curvature) hypersurfaces, together with spatially harmonic coordinates.…

We construct a uniform local bound of curvature operator from local bounds of Ricci curvature and injectivity radius among all $n$-dimensional Ricci flows. Thus new compactness theorems for the Ricci flow and Ricci solitons are derived. In particular, we show that every Ricci flow with $|Ric|\leq K$ must satisfy $|Rm|\…

Mean curvature flow with uniform bounds on curvature and its gradient