arXiv research

Study short-time existence of conformal Ricci flow on hyperbolic manifolds.

Introduces conformal Bach flow and proves its well-posedness and backward uniqueness.

Study finite singularities in G2 structure flows using Shi-type estimates.

The paper analyzes finite-time singularities in Spin(7)-structure flows using Shi-type estimates.

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|\…

The anomaly flow on a complex 3-fold is studied with integral Shi-type estimates and long-time existence conditions.

Developing a singular dimension descent method for positive scalar curvature obstructions

The paper provides gradient estimates for a parabolic equation under Finsler geometric flows.

Motivated by the quasi-local mass problem in general relativity, we apply the asymptotically flat extensions, constructed by Shi and Tam in the proof of the positivity of the Brown--York mass, to study a fill-in problem of realizing geometric data on a 2-sphere as the boundary of a compact 3-manifold of nonnegative sca…

We develop foundational theory for the Laplacian flow for closed G_2 structures which will be essential for future study. (1). We prove Shi-type derivative estimates for the Riemann curvature tensor Rm and torsion tensor T along the flow, i.e. that a bound on $Λ(x,t)=\left(|\nabla T(x,t)|_{g(t)}^2+|Rm(x,t)|_{g(t)}^2\ri…

Researchers study heat flows of $G_{2}$-structures, proving short-time existence and uniqueness.

Constructs extensions for Bartnik data to approach mass limits.

Study harmonic flow of Spin(7)-structures on compact 8-manifolds.

Generalizes rigidity of scalar curvature for convex domains.

Study bounds total mean curvature of fill-ins with scalar curvature constraints.

We construct a solution to inverse mean curvature flow on an asymptotically hyperbolic 3-manifold which does not have the convergence properties needed in order to prove a Penrose--type inequality. This contrasts sharply with the asymptotically flat case. The main idea consists in combining inverse mean curvature flow …

Study confirms conjecture about Kähler metrics on smooth minimal models.

Positive mass theorem for tori with scalar curvature bounds.

Paper studies minimal hypersurfaces and their impact on compact manifolds with nonnegative scalar curvature.

In this paper, we give the first detailed proof of the short-time existence of Deane Yang's local Ricci flow. Then using the local Ricci flow, we prove short-time existence of the Ricci flow on noncompact manifolds, whose Ricci curvature has global lower bound and sectional curvature has only local average integral bou…

Suppose that $Σ=\partial M$ is the $n$-dimensional boundary of a connected compact Riemannian spin manifold $( M,\langle\;,\;\rangle)$ with non-negative scalar curvature, and that the (inward) mean curvature $H$ of $Σ$ is positive. We show that the first eigenvalue of the Dirac operator of the boundary corresponding to…

In this paper we consider the Ricci flow on manifolds with boundary with appropriate control on its mean curvature and conformal class. We obtain higher order estimates for the curvature and second fundamental form near the boundary, similar to Shi's local derivative estimates. As an application, we prove a version of …

We give a simple proof of an extension of the existence results of Ricci flow of G.Giesen and P.M.Topping [GiT1],[GiT2], on incomplete surfaces with bounded above Gauss curvature without using the difficult Shi's existence theorem of Ricci flow on complete non-compact surfaces and the pseudolocality theorem of G.Perelm…

Study of Brown--York mass for four-dimensional asymptotically flat manifolds.

Derives heat equation estimates linked to Ricci flow on compact and noncompact manifolds.

Consider a complex analytic manifold $X$ and a coherent Lie subalgebra $\shi$ of the Lie algebra of complex vector fields on $X$. By using a natural $\shd_X$-module $\shm_\shi$ naturally associated to $\shi$ and the ring (in the derived sense) $\rhom[\shd_X](\shm_\shi,\shm_\shi)$, we associate integers which measure th…

Volume comparison theorem for rank 1 symmetric spaces proved.

Second Ricci flow proves existence of Kaehler-Einstein metrics on noncompact manifolds.

We show that the solution constructed in an earlier work of Y-G. Shi and the authors can be used to obtain sharp gradient estimates for the Kaehler-Ricci flow which achieves equality on a steady soliton. The estimate can be applied to obtain a long time existence of the Kaehler-Ricci flow. In the second part of the pap…

In this short note we present local derivative estimates for heat equations on Riemannian manifolds following the line of W.-X. Shi. As an application we generalize a second derivative estimate of R. Hamilton for heat equations on compact manifolds to noncompact case.

New flow for G2-structures helps find torsion-free structures.

Paper provides lower bounds for eigenvalues on singular Riemannian foliations.

Geometric flow on symplectic manifolds connects to Type IIA string theory.

In this note, we prove the following generalization of a theorem of Shi and Tam \cite{ShiTam02}: Let $(Ω, g)$ be an $n$-dimensional ($n \geq 3$) compact Riemannian manifold, spin when $n>7$, with non-negative scalar curvature and mean convex boundary. If every boundary component $Σ_i$ has positive scalar curvature and …

Real analyticity proved for modified Laplacian coflow solutions.

Parabolic flow techniques applied to 11D supergravity solutions.

Given a completely arbitrary surface, whether or not it has bounded curvature, or even whether or not it is complete, there exists an instantaneously complete Ricci flow evolution of that surface that exists for a specific amount of time [GT11]. In the case that the underlying Riemann surface supports a hyperbolic metr…

Enhanced Euler characteristic improves knot homology detection.

New flow preserves singularities on incomplete manifolds.

We study the supremum of the total mean curvature on the boundary of compact, mean-convex 3-manifolds with nonnegative scalar curvature, and a prescribed boundary metric. We establish an additivity property for this supremum and exhibit rigidity for maximizers assuming the supremum is attained. When the boundary consis…

Flow solves $G_2$ system, proving existence of torsion-free metrics.

Consider a triple of "Bartnik data" $(Σ, γ,H)$, where $Σ$ is a topological 2-sphere with Riemannian metric $γ$ and positive function $H$. We view Bartnik data as a boundary condition for the problem of finding a compact Riemannian 3-manifold $(Ω,g)$ of nonnegative scalar curvature whose boundary is isometric to $(Σ,γ)$…

The paper studies a gradient flow of $G_2$ structures and proves existence and convergence results.

In \cite{ly, ly2}, Liu and the second author propose a definition of the quasi-local mass and prove its positivity. This is demonstrated through an inequality which in turn can be interpreted as a total mean curvature comparison theorem for isometric embeddings of a surface of positive Gaussian curvature. The Riemannia…

In this paper, we obtain a positivity result of a quasi-local mass integral as proposed by Shi and Tam in general dimensions. The main argument is based on the monotonicity of a mass integral in a foliation of quasi-spherical metrics and a positive mass type theorem which was proved by Wang and Yau in the three dimensi…

In this paper, we continue to study the generalized Ricci flow. We give a criterion on steady gradient Ricci soliton on complete and noncompact Riemannian manifolds that is Ricci-flat, and then introduce a natural flow whose stable points are Ricci-flat metrics. Modifying the argument used by Shi and List, we prove the…

Mean curvature flow is not a gradient flow on two nondegenerate metric spaces.

The paper estimates curvature for a specific type of equations.