268 research outputs found
Regularity estimates of solutions to complex Monge-Amp\`ere equations on Hermitian manifolds
In this paper, we obtain the Bedford-Taylor interior estimate and
local Calabi estimate for the solutions to complex Monge-Amp\`ere
equations on Hermitian manifolds.Comment: 18 page
Measure Estimates, Harnack Inequalities and Ricci Lower Bound
On a Riemannian metric-measure space, we establish an
Alexandrov-Bakelman-Pucci type measure estimate connecting Bakry-\'Emery Ricci
curvature lower bound, modified Laplacian and the measure of certain special
sets. We apply this estimate to prove Harnack inequalities for the modified
Laplacian operator and fully non-linear operators. These inequalities seem not
available in the literature; And our proof, solely based on the ABP estimate,
does not involve any Sobolev inequalities nor gradient estimate. We also
propose a question regarding the characterization of Ricci lower bound by the
Harnack inequality.Comment: 37 pages. We fix an error in the proof of Lemma 9.3 in previous
version which was pointed out to us by Prof. Diego Maldonad
A rigidity theorem for surfaces in Schwarzschild manifold
In this article, we prove a rigidity theorem for isometric embeddings into
the Schwarzschild manifold, by using the variational formula of quasi-local
mass.Comment: 8 page
Regularity of Degenerate Hessian Equation
We show a second order a priori estimate for solutions to the complex
-Hessian equation on a compact K\"ahler manifold provided the --st
root of the right hand side is . This improves an estimate of
Hou-Ma-Wu. An example is provided to show that the exponent is sharp.Comment: 25 page
Fu-Yau Hessian Equations
We solve the Fu-Yau equation for arbitrary dimension and arbitrary slope
. Actually we obtain at the same time a solution of the open case
, an improved solution of the known case , and solutions
for a family of Hessian equations which includes the Fu-Yau equation as a
special case. The method is based on the introduction of a more stringent
ellipticity condition than the usual admissible cone condition, and
which can be shown to be preserved by precise estimates with scale.Comment: 36 page
The Anomaly flow on unimodular Lie groups
The Hull-Strominger system for supersymmetric vacua of the heterotic string
allows general unitary Hermitian connections with torsion and not just the
Chern unitary connection. Solutions on unimodular Lie groups exploiting this
flexibility were found by T. Fei and S.T. Yau. The Anomaly flow is a flow whose
stationary points are precisely the solutions of the Hull-Strominger system.
Here we examine its long-time behavior on unimodular Lie groups with general
unitary Hermitian connections. We find a diverse and intricate behavior, which
depends very much on the Lie group and the initial data.Comment: 24 page
A second order estimate for general complex Hessian equations
We derive a priori estimates for the -plurisubharmonic solutions
of general complex Hessian equations with right-hand side depending on
gradients.Comment: 17 pages, final version, to appear in Analysis and PD
New curvature flows in complex geometry
This is a survey of some of the recent developments on the geometric and
analytic aspects of the Anomaly flow. It is a flow of -forms on a
-fold which was originally motivated by string theory and the need to
preserve the conformally balanced property of a Hermitian metric in the absence
of a -Lemma. It has revealed itself since to be a
remarkable higher order extension of the Ricci flow. It has also led to several
other curvature flows which may be interesting from the point of view of both
non-K\"ahler geometry and the theory of non-linear partial differential
equations.Comment: References and a short section adde
On estimates for the Fu-Yau generalization of a Strominger system
We study an equation proposed by Fu and Yau as a natural -dimensional
generalization of a Strominger system that they solved in dimension . It is
a complex Hessian equation with right hand side depending on gradients.
Building on the methods of Fu and Yau, we obtain , and
a priori estimates. We also identify difficulties in extending
the Fu-Yau arguments for non-degeneracy from dimension to higher
dimensions.Comment: 34 pages, final version, to appear in Crelle's Journa
The Fu-Yau equation with negative slope parameter
The Fu-Yau equation is an equation introduced by J. Fu and S.T. Yau as a
generalization to arbitrary dimensions of an ansatz for the Strominger system.
As in the Strominger system, it depends on a slope parameter . The
equation was solved in dimension by Fu and Yau in two successive papers for
, and for . In the present paper, we solve the Fu-Yau
equation in arbitrary dimension for . To our knowledge, these are
the first non-trivial solutions of the Fu-Yau equation in any dimension
strictly greater than .Comment: 30 pages, final version, to appear in Invent. Mat
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