70 research outputs found
Torus equivariant K-stability
It is conjectured that to test the K-polystability of a polarised variety it is enough to consider test-configurations which are equivariant with respect to a torus in the automorphism group. We prove partial results towards this conjecture. We also show that it would give a new proof of the K-polystability of constant scalar curvature polarised manifolds
Asymptotic stability of the Cauchy and Jensen functional equations
The aim of this note is to investigate the asymptotic stability behaviour of
the Cauchy and Jensen functional equations. Our main results show that if these
equations hold for large arguments with small error, then they are also valid
everywhere with a new error term which is a constant multiple of the original
error term. As consequences, we also obtain results of hyperstability character
for these two functional equations
Hamiltonian 2-forms in Kahler geometry, III Extremal metrics and stability
This paper concerns the explicit construction of extremal Kaehler metrics on
total spaces of projective bundles, which have been studied in many places. We
present a unified approach, motivated by the theory of hamiltonian 2-forms (as
introduced and studied in previous papers in the series) but this paper is
largely independent of that theory.
We obtain a characterization, on a large family of projective bundles, of
those `admissible' Kaehler classes (i.e., the ones compatible with the bundle
structure in a way we make precise) which contain an extremal Kaehler metric.
In many cases, such as on geometrically ruled surfaces, every Kaehler class is
admissible. In particular, our results complete the classification of extremal
Kaehler metrics on geometrically ruled surfaces, answering several
long-standing questions.
We also find that our characterization agrees with a notion of K-stability
for admissible Kaehler classes. Our examples and nonexistence results therefore
provide a fertile testing ground for the rapidly developing theory of stability
for projective varieties, and we discuss some of the ramifications. In
particular we obtain examples of projective varieties which are destabilized by
a non-algebraic degeneration.Comment: 40 pages, sequel to math.DG/0401320 and math.DG/0202280, but largely
self-contained; partially replaces and extends math.DG/050151
Relative K-stability for KĂ€hler manifolds
We study the existence of extremal KĂ€hler metrics on KĂ€hler manifolds.
After introducing a notion of relative K-stability for Kahler manifolds,
we prove that KĂ€hler manifolds admitting extremal KĂ€hler metrics are relatively K-stable. Along the way, we prove a general Lp lower bound on the
Calabi functional involving test configurations and their associated numerical invariants, answering a question of Donaldson.
When the KÀhler manifold is projective, our definition of relative K-stability is stronger than the usual definition given by Székelyhidi. In particular our result strengthens the known results in the projective case (even for constant scalar curvature KÀhler metrics), and rules out a well known counterexample to the "naïve" version of the Yau-Tian-Donaldson conjecture in this setting
Ancient solutions and translators of Lagrangian mean curvature flow
Suppose that is an almost calibrated, exact, ancient solution
of Lagrangian mean curvature flow in . We show that if
has a blow-down given by the static union of two Lagrangian
subspaces with distinct Lagrangian angles that intersect along a line, then
is a translator. In particular in , all
zero-Maslov, exact, ancient solutions of Lagrangian mean curvature flow with
entropy less than 3 are special Lagrangian, a union of planes, or translators.Comment: 24 page
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