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Albert Chau

Publications and source records attributed to Albert Chau.

At least 37 records · Page 2Linked to original sources
arXiv2015

Longtime existence of the Kähler-Ricci flow on $\Bbb C ^n$

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-negative holomorphic bisectional curvature, and that the solution converges as $t\to \infty$ to the standard Euclidean metric after rescaling. We also prove longtime existence results for more general Kähler metrics on $\Bbb C ^n$ which are not necessarily $U(n)$-invariant.

math.DG↗
arXiv2015

Monge Ampère functionals and the second boundary value problem

We consider a Monge-Ampère functional and its corresponding second boundary value problem, a nonlinear fourth order PDE with two Dirichlet boundary conditions. This problem was solved by Trudinger-Wang and Le under the assumption that the right hand side of the equation is nonpositive. We remove this assumption, to settle the case of the second boundary value problem with arbitrary right hand side, in dimensions $n \ge 2$. In particular, this shows that one can prescribe the affine mean curvature of the graph of a convex function with Dirichlet boundary conditions on the function and the determinant of its Hessian. We relate our results, and the case of $n=1$, to a notion of properness for a certain functional on the set of convex functions.

math.AP↗
arXiv2014

Deforming complete Hermitian metrics with unbounded curvature

We produce solutions to the Kähler-Ricci flow emerging from complete initial metrics $g_0$ which are $C^0$ Hermitian limits of Kähler metrics. Of particular interest is when $g_0$ is Kähler with unbounded curvature. We provide such solutions for a wide class of $U(n)$-invariant Kähler metrics $g_0$ on $n$ dimensional complex Euclidean space, many of which having unbounded curvature. As a special case we have the following Corollary: The Kähler-Ricci flow has a smooth short time solution starting from any smooth complete $U(n)$-invariant Käbler metric on $\C^n$ with either non-negative or non-positive holomorphic bisectional curvature, and the solution exists for all time in the case of non-positive curvature.

math.DG↗
arXiv2012

Kähler C-spaces and quadratic bisectional curvature

In this article we give necessary and sufficient conditions for an irreducible Kähler C-space with $b_2=1$ to have nonnegative or positive quadratic bisectional curvature, assuming the space is not Hermitian symmetric. In the cases of the five exceptional Lie groups $E_6, E_7, E_8, F_4, G_2$, the computer package MAPLE is used to assist our calculations. The results are related to two conjectures of Li-Wu-Zheng.

math.DG↗
arXiv2011

On quadratic orthogonal bisectional curvature

In this article we study compact K\ahler manifolds satisfying a certain nonnegativity condition on the bisectional curvature. Under this condition, we show that the scalar curvature is nonnegative and that the first Chern class is positive assuming local irreducibility. We also obtain a partial classification of possible de Rham decompositions of the universal cover under this condition

math.DG↗
arXiv2011

Lagrangian Mean Curvature flow for entire Lipschitz graphs II

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 large supercritical Lagrangian phase condition.

math.DG↗
arXiv2010

Rigidity of Entire self-shrinking solutions to curvature flows

We show that (a) any entire graphic self-shrinking solution to the Lagrangian mean curvature flow in ${\mathbb C}^{m}$ with the Euclidean metric is flat; (b) any space-like entire graphic self-shrinking solution to the Lagrangian mean curvature flow in ${\mathbb C}^{m}$ with the pseudo-Euclidean metric is flat if the Hessian of the potential is bounded below quadratically; and (c) the Hermitian counterpart of (b) for the Kähler Ricci flow.

math.DG↗
arXiv2009

On a modified parabolic complex Monge-Ampère equation with applications

We study a parabolic complex Monge-Ampère type equation of the form \eqref{MA} on a complete noncompact \K manifold. We prove a short time existence result and obtain basic estimates. Applying these results, we prove that under certain assumptions on a given real and closed (1,1) form $Ω$ and initial \K metric $g_0$ on $M$, the modified \KR flow $g'=-\Ric+Ω$ has a long time smooth solution converging to a complete \K metric such that $\Ric=Ω$, which extends the result in [1] to non-compact manifolds. We will also obtain a long time existence result for the \KR flow which generalizes a result [5].

math.DG↗
arXiv2009

Entire self-similar solutions to Lagrangian Mean curvature flow

We consider self-similar solutions to mean curvature evolution of entire Lagrangian graphs. When the Hessian of the potential function $u$ has eigenvalues strictly uniformly between -1 and 1, we show that on the potential level all the shrinking solitons are quadratic polynomials while the expanding solitons are in one-to-one correspondence to functions of homogenous of degree 2 with the Hessian bound. We also show that if the initial potential function is cone-like at infinity then the scaled flow converges to an expanding soliton as time goes to infinity.

math.DG↗
arXiv2009

Lagrangian Mean Curvature flow for entire Lipschitz graphs

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 smooth for all positive time and satisfies uniform estimates away from time $t=0$. In particular, under the mean curvature flow the graph immediately becomes smooth and the solution exists for all time such that the second fundamental form decays uniformly to 0 on the graph as $t\to \infty$. Our assumption on the Lipschitz norm is equivalent to the assumption that the underlying Lagrangian potential $u$ is uniformly convex with its Hessian bounded in $L^\infty$. We apply this result to prove a Bernstein type theorem for translating solitons, namely that if such an entire Lagrangian graph is a smooth translating soliton, then it must be a flat plane. We also prove convergence of the evolving graphs under additional conditions.

math.DG↗
arXiv2009

A $C^0$-estimate for the parabolic Monge-Ampère equation on complete non-compact Kähler manifolds

In this article we study the Kähler Ricci flow, the corresponding parabolic Monge Ampère equation and complete non-compact Kähler Ricci flat manifolds. In our main result Theorem \ref{mainthm} we prove that if $(M, g)$ is sufficiently close to being Kähler Ricci flat in a suitable sense, then the Kähler Ricci flow \eqref{KRF} has a long time smooth solution $g(t)$ converging smoothly uniformly on compact sets to a complete Kähler Ricci flat metric on $M$. The main step is to obtain a uniform $C^0$-estimates for the corresponding parabolic Monge Ampère equation. Our results on this can be viewed as a parabolic version of the main results in \cite{TY3} on the elliptic Monge Ampère equation.

math.DG↗
arXiv2008

On the simply connectedness of non-negatively curved Kähler manifolds and applications

We study complete noncompact long time solutions $(M, g(t))$ to the Kähler-Ricci flow with uniformly bounded nonnegative holomorphic bisectional curvature. We will show that when the Ricci curvature is positive and uniformly pinched, i.e. $ R_\ijb \ge cRg_\ijb$ at $(p,t)$ for all $t$ for some $c>0$, then there always exists a local gradient Kähler Ricci soliton limit around $p$ after possibly rescaling $g(t)$ along some sequence $t_i \to \infty$. We will show as an immediate corollary that the injectivity radius of $g(t)$ along $t_i$ is uniformly bounded from below along $t_i$, and thus $M$ must in fact be simply connected. Additional results concerning the uniformization of $M$ and fixed points of the holomorphic isometry group will also be established. We will then consider removing the condition of positive Ricci for $(M, g(t))$. Combining our results with Cao's splitting for Kähler Ricci flow \cite{Cao04} and techniques of Ni-Tam \cite{NiTam03}, we show that when the positive eigenvalues of the Ricci curvature are uniformly pinched at some point $p \in M$, then $M$ has a special holomorphic fiber bundle structure. We will treat a special cases, complete Kähler manifolds with non-negative holomorphic bisectional and average quadratic curvature decay as well as the case of steady gradient Kähler Ricci solitons.

math.DG↗
arXiv2007

A survey on the Kähler-Ricci flow and Yau's uniformization conjecture

Yau's uniformization conjecture states: a complete noncompact Kähler manifold with positive holomorphic bisectional curvature is biholomorphic to $\ce^n$. The Kähler-Ricci flow has provided a powerful tool in understanding the conjecture, and has been used to verify the conjecture in several important cases. In this article we present a survey of the Kähler-Ricci flow with focus on its application to uniformization. Other interesting methods and results related to the study of Yau's conjecture are also discussed.

math.DG↗
arXiv2007

On the Steinness of a class of Kähler manifolds

Let $(M^n, g)$ be a complete non-compact Kähler manifold with non-negative and bounded holomorphic bisectional curvature. We prove that $M$ is holomorphically covered by a pseudoconvex domain in $\C^n$ which is homeomorphic to $\R^{2n}$, provided $(M^n, g)$ has uniform linear average quadratic curvature decay.

math.DG↗
arXiv2007

Pseudolocality for the Ricci flow and applications

In \cite{P1}, Perelman established a differential Li-Yau-Hamilton (LYH) type inequality for fundamental solutions of the conjugate heat equation corresponding to the Ricci flow on compact manifolds (also see \cite{N2}). As an application of the LYH inequality, Perelman proved a pseudolocality result for the Ricci flow on compact manifolds. In this article we provide the details for the proofs of these results in the case of a complete non-compact Riemannian manifold. Using these results we prove that under certain conditions, a finite time singularity of the Ricci flow must form within a compact set. We also prove a long time existence result for the \KRF flow on complete non-negatively curved \K manifolds.

math.DG↗
arXiv2005

Non-negatively curved Kähler manifolds with average quadratic curvature decay

Let $(M, g)$ be a complete non-compact Kähler manifold with non-negative and bounded holomorphic bisectional curvature. Extending our techniques developed in \cite{CT3}, we prove that the universal cover $\wt M$ of $M$ is biholomorphic to $\ce^n$ provided either that $(M, g)$ has average quadratic curvature decay, or $M$ supports an eternal solution to the Kähler-Ricci flow with non-negative and uniformly bounded holomorphic bisectional curvature. We also classify certain local limits arising from the Kähler-Ricci flow in the absence of uniform estimates on the injectivity radius.

math.DG↗
arXiv2005

On the complex structure of Kähler manifolds with nonnegative curvature

We study the asymptotic behavior of the Kähler-Ricci flow on Kähler manifolds of nonnegative holomorphic bisectional curvature. Using these results we prove that a complete noncompact Kähler manifold with nonnegative bounded holomorphic bisectional curvature and maximal volume growth is biholomorphic to complex Euclidean space $\C^n$. We also show that the volume growth condition can be removed if we assume $(M, g)$ has average quadratic scalar curvature decay (see Theorem 2.1) and positive curvature operator.

math.DG↗