446 research outputs found
Uniform Sobolev inequality along the Sasaki-Ricci flow
We prove a uniform Sobolev inequality along the Sasaki-Ricci flow. In the
process, we develop the theory of basic Lebesgue and Sobolev function spaces,
and prove some general results about the decomposition of the heat kernel for a
class of elliptic operators on a Sasaki manifold.Comment: 13 page
The Transverse Entropy Functional and the Sasaki-Ricci Flow
We introduce two new functionals on Sasaki manifolds, inspired by the work of
Perelman, which are monotonic along the Sasaki-Ricci flow. We relate their
gradient flow, via diffeomorphisms preserving the foliated structure of the
manifold, to the transverse Ricci flow. Finally, when the basic first Chern
class is positive, we employ these new functionals to prove a uniform
bound for the transverse scalar curvature, and a uniform bound for the
transverse Ricci potential along the Sasaki-Ricci flow.Comment: 24 pages. Minor changes incorporating the referee's suggestions.
Final version, to appear in Trans. of the AM
estimates for nonlinear elliptic equations of twisted type
We prove a priori interior estimates for solutions of fully
nonlinear elliptic equations of twisted type. For example, our estimates apply
to equations of the type convex + concave. These results are particularly well
suited to equations arising from elliptic regularization. As application, we
obtain a new proof of an estimate of Streets-Warren on the twisted real
Monge-Ampere equation
Stability and Convergence of the Sasaki-Ricci Flow
We introduce a holomorphic sheaf E on a Sasaki manifold and study two new
notions of stability for E along the Sasaki-Ricci flow related to the `jumping
up' of the number of global holomorphic sections of E at infinity. First, we
show that if the Mabuchi K-energy is bounded below, the transverse Riemann
tensor is bounded in C^{0} along the flow, and the C -infinity closure of the
Sasaki structure under the diffeomorphism group does not contain a Sasaki
structure with strictly more global holomorphic sections of E, then the
Sasaki-Ricci flow converges exponentially fast to a Sasaki-Einstein metric.
Secondly, we show that if the Futaki invariant vanishes, and the lowest
positive eigenvalue of the d-bar Laplacian on global sections of E is bounded
away from zero uniformly along the flow, then the Sasaki-Ricci flow converges
exponentially fast to a Sasaki-Einstein metric.Comment: 20 page
Remarks on the Yang-Mills flow on a compact Kahler manifold
We study the Yang-Mills flow on a holomorphic vector bundle E over a compact
Kahler manifold X. We construct a natural barrier function along the flow, and
introduce some techniques to study the blow-up of the curvature along the flow.
Making some technical assumptions, we show how our techniques can be used to
prove that the curvature of the evolved connection is uniformly bounded away
from an analytic subvariety determined by the Harder-Narasimhan-Seshadri
filtration of E. We also discuss how our assumptions are related to stability
in some simple cases.Comment: We found a mistake in the proof of Proposition 4. The paper has been
completely rewritten. The title and abstract have been altered to reflect the
changes. 26 page
A singular Demailly-Paun theorem
We give a numerical characterization of the Kahler cone of a possibly
singular compact analytic variety which is embedded in a smooth ambient space.Comment: 7 page
Convergence of the -flow on toric manifolds
We show that on a Kahler manifold whether the J-flow converges or not is
independent of the chosen background metric in its Kahler class. On toric
manifolds we give a numerical characterization of when the J-flow converges,
verifying a conjecture of Lejmi and the second author in this case. We also
strengthen existing results on more general inverse equations on
Kahler manifolds.Comment: 28 page
Dimension of the minimum set for the real and complex Monge-Amp\`{e}re equations in critical Sobolev spaces
We prove that the zero set of a nonnegative plurisubharmonic function that
solves in and is
in contains no analytic sub-variety of dimension
or larger. Along the way we prove an analogous result for the real
Monge-Amp\`ere equation, which is also new. These results are sharp in view of
well-known examples of Pogorelov and B{\l}ocki. As an application, in the real
case we extend interior regularity results to the case that lies in a
critical Sobolev space (or more generally, certain Sobolev-Orlicz spaces).Comment: 10 page
K-Semistability for irregular Sasakian manifolds
We introduce a notion of K-semistability for Sasakian manifolds. This extends
to the irregular case the orbifold K-semistability of Ross-Thomas. Our main
result is that a Sasakian manifold with constant scalar curvature is
necessarily K-semistable. As an application, we show how one can recover the
volume minimization results of Martelli-Sparks-Yau, and the Lichnerowicz
obstruction of Gauntlett-Martelli-Sparks-Yau from this point of view.Comment: 25 page
An extension theorem for K\"ahler currents
We prove an extension theorem for Kahler currents with analytic singularities
in a Kahler class on a complex submanifold of a compact Kahler manifold.Comment: 12 pages, 2 figures; corrected typos, final version to appear in Ann.
Fac. Sci. Toulouse Mat
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