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Duong H. Phong

Publications and source records attributed to Duong H. Phong.

At least 19 recordsLinked to original sources
arXiv2025

Geometric invariants and the Monge-Ampere equation in Kähler geometry

This is a contribution to the special issue of Surveys in Differential Geometry celebrating the 75th birthday of Shing-Tung Yau. The bulk of the paper is devoted to a survey of some new geometric inequalities and estimates for the Monge-Ampere equation, obtained by the authors in the last few years in joint work with F. Tong, J. Song, and J. Sturm. These all depend in an essential way on Yau's solution of the Calabi conjecture, which is itself nearing its own 50th birthday. The opportunity is also taken to survey briefly many current directions in complex geometry, which he more recently pioneered.

math.DG↗
arXiv2024

Diameter estimates in Kähler geometry II: removing the small degeneracy assumption

In this short note, we remove the small degeneracy assumption in our earlier works [10, 11]. This is achieved by a technical improvement of Corollary 5.1 in [10]. As a consequence, we establish the same geometric estimates for diameter, Green's functions and Sobolev inequalities under an entropy bound for the Kähler metrics, without any small degeneracy assumption.

math.DG↗
arXiv2024

Uniform $L^\infty$ estimates: subsolutions to fully nonlinear partial differential equations

Uniform bounds are obtained using the auxiliary Monge-Ampère equation method for solutions of very general classes of fully non-linear partial differential equations, assuming the existence of a ${C}$-subsolution in the sense of G. Székelyhidi and B. Guan. The main advantage over previous estimates is in the Kähler case, where the new estimates remain uniform as the background metric is allowed to degenerate.

math.AP↗
arXiv2023

Sobolev inequalities on Kähler spaces

We establish a uniform Sobolev inequality for Kähler metrics, which only require an entropy bound and no lower bound on the Ricci curvature. We further extend our Sobolev inequality to singular Kähler metrics on Kähler spaces with normal singularities. This allows us to build a general theory of global geometric analysis on singular Kähler spaces including the spectral theorem, heat kernel estimates, eigenvalue estimates and diameter estimates. Such estimates were only known previously in very special cases such as Bergman metrics. As a consequence, we derive various geometric estimates, such as the diameter estimate and the Sobolev inequality, for Kähler-Einstein currents on projective varieties with definite or vanishing first Chern class.

math.DG↗
arXiv2023

Geometric flows from unified string theories

A survey of new geometric flows motivated by string theories is provided. Their settings can range from complex geometry to almost-complex geometry to symplectic geometry. From the PDE viewpoint, many of them can be viewed as intermediate flows between the Ricci flow and the Kähler-Ricci flow, albeit often coupled to flows of additional fields. In particular, a survey is given of joint works of the author with Tristan Collins, Teng Fei, Bin Guo, Sebastien Picard, and Xiangwen Zhang.

math.DG↗
arXiv2023

On $L^\infty$ estimates for fully nonlinear partial differential equations

Sharp $L^\infty$ estimates are obtained for general classes of fully non-linear PDE's on non-Kähler manifolds, complementing the theory developed earlier by the authors in joint work with F. Tong for the Kähler case. The key idea is still a comparison with an auxiliary Monge-Ampère equation, but this time on a ball with Dirichlet boundary conditions, so that it always admits a unique solution. The method applies not just to compact Hermitian manifolds, but also to the Dirichlet problem, to open manifolds with a positive lower bound on their injectivity radii, to $(n-1)$ form equations, and even to non-integrable almost-complex or symplectic manifolds. It is the first method applicable in any generality to large classes of non-linear equations, and it usually improves on other methods when they happen to be available for specific equations.

math.DG↗
arXiv2022

Auxiliary Monge-Ampere equations in geometric analysis

This is an introduction to a particular class of auxiliary complex Monge-Ampère equations which had been instrumental in $L^\infty$ estimates for fully non-linear equations and various questions in complex geometry. The essential comparison inequalities are reviewed and shown to apply in many contexts. Adapted to symplectic geometry, with the auxiliary equation given now by a real Monge-Ampère equation, the method gives an improvement of an earlier theorem of Tosatti-Weinkove-Yau, reducing Donaldson's conjecture on the Calabi-Yau equation with a taming symplectic form from an exponential bound to an $L^1$ bound.

math.DG↗
arXiv2022

Diameter estimates in Kähler geometry

Diameter estimates for Kähler metrics are established which require only an entropy bound and no lower bound on the Ricci curvature. The proof builds on recent PDE techniques for $L^\infty$ estimates for the Monge-Ampère equation, with a key improvement allowing degeneracies of the volume form of codimension strictly greater than one. As a consequence, diameter bounds are obtained for long-time solutions of the Kähler-Ricci flow and finite-time solutions when the limiting class is big, as well as for special fibrations of Calabi-Yau manifolds.

math.DG↗
arXiv2022

Uniform entropy and energy bounds for fully non-linear equations

Energy bounds which are uniform in the background metric are obtained from upper bounds for entropy-like quantities. The argument is based on auxiliary Monge-Ampère equations involving sublevel sets, and bypasses the Alexandrov-Bakelman-Pucci maximum principle. In particular, it implies uniform $L^\infty$ bounds for systems coupling a fully non-linear equation to its linearization, generalizing the cscK equation.

math.DG↗
arXiv2022

Green's functions and complex Monge-Ampère equations

Uniform $L^1$ and lower bounds are obtained for the Green's function on compact Kähler manifolds. Unlike in the classic theorem of Cheng-Li for Riemannian manifolds, the lower bounds do not depend directly on the Ricci curvature, but only on integral bounds for the volume form and certain of its derivatives. In particular, a uniform lower bound for the Green's function on Kähler manifolds is obtained which depends only on a lower bound for the scalar curvature and on an $L^q$ norm for the volume form for some $q>1$. The proof relies on auxiliary Monge-Ampère equations, and is fundamentally non-linear. The lower bounds for the Green's function imply in turn $C^1$ and $C^2$ estimates for complex Monge-Ampère equations with a sharper dependence on the function on the right hand side.

math.DG↗
arXiv2022

Spinor flows with flux, I: short-time existence and smoothing estimates

Spinor fields which are covariantly constant with respect to a connection with flux are of particular interest in unified string theories and supergravity theories, as their existence is required by supersymmetry. In this paper, flows of spinor fields are introduced which are parabolic and whose stationary points are such covariantly constant spinors.

math.DG↗
arXiv2021

Stability of the Type IIA flow and its applications in symplectic geometry

In this paper the dynamical stability of the Type IIA flow with no source near its stationary points is established. These stationary points had been shown previously by the authors to be Ricci-flat Kähler metrics on Calabi-Yau 3-folds. The dynamical stability of the Type IIA flow is then applied to prove the stability under symplectic deformations of the Kähler property for Calabi-Yau 3-folds.

math.DG↗
arXiv2021

On the modulus of continuity of solutions to complex Monge-Ampère equations

In this paper, we prove a uniform and sharp estimate for the modulus of continuity of solutions to complex Monge-Ampère equations, using the PDE-based approach developed by the first three authors in their approach to supremum estimates for fully non-linear equations in Kähler geometry. As an application, we derive a uniform diameter bound for Kähler metrics satisfying certain Monge-Ampère equations.

math.DG↗
arXiv2021

Symplectic geometric flows

Several geometric flows on symplectic manifolds are introduced which are potentially of interest in symplectic geometry and topology. They are motivated by the Type IIA flow and T-duality between flows in symplectic geometry and flows in complex geometry. Examples include the Hitchin gradient flow on symplectic manifolds, and a new flow which is called the dual Ricci flow.

math.SG↗
arXiv2021

On $L^\infty$ estimates for Monge-Ampère and Hessian equations on nef classes

The PDE approach developed earlier by the first three authors for $L^\infty$ estimates for fully non-linear equations on Kähler manifolds is shown to apply as well to Monge-Ampère and Hessian equations on nef classes. In particular, one obtains a new proof of the estimates of Boucksom-Eyssidieux-Guedj-Zeriahi and Fu-Guo-Song for the Monge-Ampère equation, together with their generalization to Hessian equations.

math.DG↗
arXiv2021

Stability estimates for the complex Monge-Ampère and Hessian equations

A new proof for stability estimates for the complex Monge-Ampère and Hessian equations is given, which does not require pluripotential theory. A major advantage is that the resulting stability estimates are then uniform under general degenerations of the background metric in the case of the Monge-Ampère equation, and under degenerations to a big class in the case of Hessian equations.

math.DG↗
arXiv2021

On $L^\infty$ estimates for complex Monge-Ampère equations

A PDE proof is provided for the sharp $L^\infty$ estimates for the complex Monge-Ampère equation which had required pluripotential theory before. The proof covers both cases of fixed background as well as degenerating background metrics. It extends to more general fully non-linear equations satisfying a structural condition, and it also gives estimates of Trudinger type.

math.DG↗