666 research outputs found
Hitchhiker's guide to the fractional Sobolev spaces
This paper deals with the fractional Sobolev spaces W^[s,p]. We analyze the
relations among some of their possible definitions and their role in the trace
theory. We prove continuous and compact embeddings, investigating the problem
of the extension domains and other regularity results. Most of the results we
present here are probably well known to the experts, but we believe that our
proofs are original and we do not make use of any interpolation techniques nor
pass through the theory of Besov spaces. We also present some counterexamples
in non-Lipschitz domains
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
L^1 metric geometry of big cohomology classes
Suppose is a compact K\"ahler manifold of dimension , and
is closed -form representing a big cohomology class. We
introduce a metric on the finite energy space ,
making it a complete geodesic metric space. This construction is potentially
more rigid compared to its analog from the K\"ahler case, as it only relies on
pluripotential theory, with no reference to infinite dimensional Finsler
geometry. Lastly, by adapting the results of Ross and Witt Nystr\"om to the big
case, we show that one can construct geodesic rays in this space in a flexible
manner
Monge-Ampère measures on contact sets
Let be a compact K\"ahler manifold of complex dimension n and
be a smooth closed real -form on such that its cohomology
class is pseudoeffective. Let
be a -psh function, and let be a continuous function on
with bounded distributional laplacian with respect to such that
Then the non-pluripolar measure satisfies the equality: where, for a
subset , is the characteristic function. In
particular we prove that \[ \theta_{P_{\theta}(f)}^n= { \bf
{1}}_{\{P_{\theta}(f) = f\}} \ \theta_f^n\qquad {\rm and }\qquad
\theta_{P_\theta[\varphi](f)}^n = { \bf {1}}_{\{P_\theta[\varphi](f) = f \}} \
\theta_f^n. \
Uniqueness and short time regularity of the weak K\"ahler-Ricci flow
Let be a compact K\"ahler manifold. We prove that the K\"ahler-Ricci flow
starting from arbitrary closed positive -currents is smooth outside some
analytic subset. This regularity result is optimal meaning that the flow has
positive Lelong numbers for short time if the initial current does. We also
prove that the flow is unique when starting from currents with zero Lelong
numbers.Comment: 33 page
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