334 research outputs found
From Monge-Ampere equations to envelopes and geodesic rays in the zero temperature limit
Let X be a compact complex manifold equipped with a smooth (but not
necessarily positive) closed form theta of one-one type. By a well-known
envelope construction this data determines a canonical theta-psh function u
which is not two times differentiable, in general. We introduce a family of
regularizations of u, parametrized by a positive number beta, defined as the
smooth solutions of complex Monge-Ampere equations of Aubin-Yau type. It is
shown that, as beta tends to infinity, the regularizations converge to the
envelope u in the strongest possible Holder sense. A generalization of this
result to the case of a nef and big cohomology class is also obtained. As a
consequence new PDE proofs are obtained for the regularity results for
envelopes in [14] (which, however, are weaker than the results in [14] in the
case of a non-nef big class). Applications to the regularization problem for
quasi-psh functions and geodesic rays in the closure of the space of Kahler
metrics are given. As briefly explained there is a statistical mechanical
motivation for this regularization procedure, where beta appears as the inverse
temperature. This point of view also leads to an interpretation of the
regularizations as transcendental Bergman metrics.Comment: 28 pages. Version 2: 29 pages. Improved exposition, references
updated. Version 3: 31 pages. A direct proof of the bound on the
Monge-Amp\`ere mass of the envelope for a general big class has been included
and Theorem 2.2 has been generalized to measures satisfying a
Bernstein-Markov propert
The complex Monge-Amp\`{e}re equation on some compact Hermitian manifolds
We consider the complex Monge-Amp\`{e}re equation on compact manifolds when
the background metric is a Hermitian metric (in complex dimension two) or a
kind of Hermitian metric (in higher dimensions). We prove that the Laplacian
estimate holds when is in for any . As an
application, we show that, up to scaling, there exists a unique classical
solution in for the complex Monge-Amp\`{e}re equation when is
in .Comment: 16 pages; main result improve
On the singularity type of full mass currents in big cohomology classes
Let be a compact K\"ahler manifold and be a big cohomology
class. We prove several results about the singularity type of full mass
currents, answering a number of open questions in the field. First, we show
that the Lelong numbers and multiplier ideal sheaves of
-plurisubharmonic functions with full mass are the same as those of the
current with minimal singularities. Second, given another big and nef class
, we show the inclusion Third, we characterize big classes whose full
mass currents are "additive". Our techniques make use of a characterization of
full mass currents in terms of the envelope of their singularity type. As an
essential ingredient we also develop the theory of weak geodesics in big
cohomology classes. Numerous applications of our results to complex geometry
are also given.Comment: v2. Theorem 1.1 updated to include statement about multiplier ideal
sheaves. Several typos fixed. v3. we make our arguments independent of the
regularity results of Berman-Demaill
Pluricomplex Green's functions and Fano manifolds
We show that if a Fano manifold does not admit Kahler-Einstein metrics then
the Kahler potentials along the continuity method subconverge to a function
with analytic singularities along a subvariety which solves the homogeneous
complex Monge-Ampere equation on its complement, confirming an expectation of
Tian-Yau.Comment: EpiGA Volume 3 (2019), Article Nr.
A generalised comparison principle for the Monge-Amp\`ere equation and the pressure in 2D fluid flows
We extend the generalised comparison principle for the Monge-Amp\`ere
equation due to Rauch & Taylor (Rocky Mountain J. Math. 7, 1977) to nonconvex
domains. From the generalised comparison principle we deduce bounds (from above
and below) on solutions of the Monge-Amp\`ere equation with sign-changing
right-hand side. As a consequence, if the right-hand side is nonpositive (and
does not vanish almost everywhere) then the equation equipped with constant
boundary condition has no solutions. In particular, due to a connection between
the two-dimensional Navier-Stokes equations and the Monge-Amp\`ere equation,
the pressure in 2D Navier-Stokes equations on a bounded domain cannot
satisfy in unless (at any fixed
time). As a result at any time there exists such that
.Comment: 15 pages, 2 figure
Real Monge-Ampere equations and Kahler-Ricci solitons on toric log Fano varieties
We show, using a direct variational approach, that the second boundary value
problem for the Monge-Amp\`ere equation in R^n with exponential non-linearity
and target a convex body P is solvable iff 0 is the barycenter of P. Combined
with some toric geometry this confirms, in particular, the (generalized)
Yau-Tian-Donaldson conjecture for toric log Fano varieties (X,D), saying that
(X,D) admits a (singular) K\"ahler-Einstein metric iff it is K-stable in the
algebro-geometric sense. We thus obtain a new proof and extend to the log Fano
setting the seminal result of Zhou-Wang concerning the case when X is smooth
and D is trivial. Li's toric formula for the greatest lower bound on the Ricci
curvature is also generalized. More generally, we obtain K\"ahler-Ricci
solitons on any log Fano variety and show that they appear as the large time
limit of the K\"ahler-Ricci flow. Furthermore, using duality, we also confirm a
conjecture of Donaldson concerning solutions to Abreu's boundary value problem
on the convex body P. in the case of a given canonical measure on the boundary
of P.Comment: 53 page
- …