124 citations · 1.4k across the 62 of their papers we have counts for
47 papers · 1 filter
The Chern-Ricci flow
Valentino Tosatti, Ben Weinkove
We give a survey on the Chern-Ricci flow, a parabolic flow of Hermitian metrics on complex manifolds. We emphasize open problems and new directions.
The complex Monge-Ampère equation with a gradient term
Valentino Tosatti, Ben Weinkove
We consider the complex Monge-Ampère equation with an additional linear gradient term inside the determinant. We prove existence and uniqueness of solutions to this equation on com…
The continuity equation, Hermitian metrics and elliptic bundles
Morgan Sherman, Ben Weinkove
We extend the continuity equation of La Nave-Tian to Hermitian metrics and establish its interval of maximal existence. The equation is closely related to the Chern-Ricci flow, and…
regularity for degenerate complex Monge-Ampère equations and geodesic rays
Jianchun Chu, Valentino Tosatti, Ben Weinkove
We prove a estimate for solutions of complex Monge-Ampère equations on compact Kähler manifolds with possibly nonempty boundary, in a degenerate cohomology class. This st…
On the regularity of geodesics in the space of Kähler metrics
Jianchun Chu, Valentino Tosatti, Ben Weinkove
We prove that any two Kahler potentials on a compact Kahler manifold can be connected by a geodesic segment of C^{1,1} regularity. This follows from an a priori interior real Hessi…
The Aleksandrov-Bakelman-Pucci estimate and the Calabi-Yau equation
Valentino Tosatti, Ben Weinkove
We give two applications of the Aleksandrov-Bakelman-Pucci estimate to the Calabi-Yau equation on symplectic four-manifolds. The first is solvability of the equation on the Kodaira…
Geometric convergence of the Kähler-Ricci flow on complex surfaces of general type
Bin Guo, Jian Song, Ben Weinkove
We show that on smooth minimal surfaces of general type, the Kähler-Ricci flow starting at any initial Kähler metric converges in the Gromov-Hausdorff sense to a Kähler-Einstein or…
Gauduchon metrics with prescribed volume form
Gábor Székelyhidi, Valentino Tosatti, Ben Weinkove
We prove that on any compact complex manifold one can find Gauduchon metrics with prescribed volume form. This is equivalent to prescribing the Chern-Ricci curvature of the metrics…
The Kähler-Ricci flow on compact Kähler manifolds
Ben Weinkove
These notes are based on a lecture series given at the Park City Math Institute in the summer of 2013. The notes are intended as a leisurely introduction to the Kähler-Ricci flow o…
Inoue surfaces and the Chern-Ricci flow
Shouwen Fang, Valentino Tosatti, Ben Weinkove +1
We investigate the Chern-Ricci flow, an evolution equation of Hermitian metrics, on Inoue surfaces. These are non-Kahler compact complex surfaces of type Class VII. We show that, a…
The Kähler-Ricci flow, Ricci-flat metrics and collapsing limits
Valentino Tosatti, Ben Weinkove, Xiaokui Yang
We investigate the Kahler-Ricci flow on holomorphic fiber spaces whose generic fiber is a Calabi-Yau manifold. We establish uniform metric convergence to a metric on the base, away…
estimates for nonlinear elliptic equations in complex and almost complex geometry
Valentino Tosatti, Yu Wang, Ben Weinkove +1
We describe how to use the perturbation theory of Caffarelli to prove Evans-Krylov type estimates for solutions of nonlinear elliptic equations in complex geometry, assum…
Hermitian metrics, (n-1, n-1) forms and Monge-Ampère equations
Valentino Tosatti, Ben Weinkove
We show existence of unique smooth solutions to the Monge-Ampere equation for (n-1)-plurisubharmonic functions on Hermitian manifolds, generalizing previous work of the authors. As…
The degenerate J-flow and the Mabuchi energy on minimal surfaces of general type
Jian Song, Ben Weinkove
We prove existence, uniqueness and convergence of solutions of the degenerate J-flow on Kahler surfaces. As an application, we establish the properness of the Mabuchi energy for Ka…
The Monge-Ampère equation for (n-1)-plurisubharmonic functions on a compact Kähler manifold
Valentino Tosatti, Ben Weinkove
A C^2 function on C^n is called (n-1)-plurisubharmonic in the sense of Harvey-Lawson if the sum of any n-1 eigenvalues of its complex Hessian is nonnegative. We show that the assoc…
Collapsing of the Chern-Ricci flow on elliptic surfaces
Valentino Tosatti, Ben Weinkove, Xiaokui Yang
We investigate the Chern-Ricci flow, an evolution equation of Hermitian metrics generalizing the Kahler-Ricci flow, on elliptic bundles over a Riemann surface of genus greater than…
Local Calabi and curvature estimates for the Chern-Ricci flow
Morgan Sherman, Ben Weinkove
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.
Lecture notes on the Kähler-Ricci flow
Jian Song, Ben Weinkove
These lecture notes give an introduction to the Kahler-Ricci flow. They are based on lectures given by the authors at the conference "Analytic Aspects of Complex Algebraic Geometry…
The Chern-Ricci flow on complex surfaces
Valentino Tosatti, Ben Weinkove
The Chern-Ricci flow is an evolution equation of Hermitian metrics by their Chern-Ricci form, first introduced by Gill. Building on our previous work, we investigate this flow on c…
The J-flow on Kahler surfaces: a boundary case
Hao Fang, Mijia Lai, Jian Song +1
We study the J-flow on Kahler surfaces when the Kahler class lies on the boundary of the open cone for which global smooth convergence holds, and satisfies a nonnegativity conditio…
On the evolution of a Hermitian metric by its Chern-Ricci form
Valentino Tosatti, Ben Weinkove
We consider the evolution of a Hermitian metric on a compact complex manifold by its Chern-Ricci form. This is an evolution equation first studied by M. Gill, and coincides with th…
The Kähler-Ricci flow on projective bundles
Jian Song, Gábor Székelyhidi, Ben Weinkove
We study the behaviour of the Kähler-Ricci flow on projective bundles. We show that if the initial metric is in a suitable Kähler class, then the fibers collapse in finite time and…
Interior derivative estimates for the Kähler-Ricci flow
Morgan Sherman, Ben Weinkove
We give a maximum principle proof of interior derivative estimates for the Kähler-Ricci flow, assuming local uniform bounds on the metric.
Plurisubharmonic functions and nef classes on complex manifolds
Valentino Tosatti, Ben Weinkove
We prove the existence of plurisubharmonic functions with prescribed logarithmic singularities on complex 3-folds equipped with a nef class of positive volume. We prove the same re…