### GABOR SZEKELYHIDI THESIS

In this thesis we study the relationship between the existence of canonical metrics on a complex manifold and stability in the sense of geometric invariant theory. This can be interpreted as prescribing the scalar curvature of a torus invariant metric on an Abelian variety. Remark on the Calabi flow with bounded curvature [ abstract ] [ pdf ] Univ. We give a new proof of Bian-Guan’s constant rank theorem for nonlinear elliptic equations. We also show that the Calabi flow starting from a metric with suitable symmetry gives such a minimising sequence.

Finally we prove a general existence result on complex tori. We show that if a polarised manifold admits an extremal metric then it is K-polystable relative to a maximal torus of automorphisms. Our main result is that each tangent cone is homeomorphic to a normal affine variety. We solve Abreu’s equation with periodic right hand side, in any dimension. We give two applications of this result.

We introduce a strengthening of K-stability, based on filtrations of the homogeneous coordinate ring. The main ingredient is a result that says that a sufficiently small perturbation of a cscK manifold admits a cscK metric if it is K-polystable. We first discuss the Yau-Tian-Donaldson conjecture relating the existence of extremal metrics to an algebro-geometric stability notion and we give some example settings where this conjecture has been established.

We then give different situations in which the condition on the Mabuchi energy holds.

## Mathematics Genealogy Project

Remark on the Calabi flow with bounded curvature [ abstract ] [ pdf ] Univ. We study this discrepancy from the point of view of log K-stability. As well as improving some previous results on the behaviour of the ADM mass under the Ricci flow, we extend the analysis of the zero mass case to higher dimensions.

I am interested in geometric analysis and complex differential geometry. Fully non-linear elliptic equations on compact Hermitian manifolds [ tyesis ] [ pdf ] J.

We extend their approach to the setting where only a lower bound for the Ricci curvature is assumed. In particular we show that the maximal possible cone angle is in general smaller than the invariant R M. Optimal test-configurations for toric varieties [ abstract ] [ pdf ] J.

On a ruled surface we compute the infimum thfsis the Calabi functional for the unstable polarisations, exhibiting a decomposition analogous to the Harder-Narasimhan filtration of an unstable vector bundle.

We show that if this is piecewise linear then it gives rise to a decomposition into semistable pieces analogous to the Harder-Narasimhan filtration of an unstable vector bundle.

# Székelyhidi : On blowing up extremal Kähler manifolds

Lejmi The J-flow and stability [ abstract ] [ pdf ] Advances in Math. Finally we prove a general existence result on complex tori.

Our main result is that a Sasakian manifold with constant scalar curvature is necessarily K-semistable. The Calabi functional on a ruled surface [ abstract ] [ pdf ] Szekelyhid. We prove the conjecture stated in [32], and we relate this result to the K-stability of blown up manifolds.

Our main result is that each tangent cone is homeomorphic to a normal affine variety. We give a result in geometric invariant theory that motivates this conjecture, and an example computation that supports it.

# [math/] Extremal metrics and K-stability (PhD thesis)

In this case the infimum of the Calabi functional gzbor given by the supremum of the normalised Futaki invariants over all destabilising test-configurations, as predicted by a conjecture of Donaldson. Collins Convergence of the J-flow on toric manifolds [ abstract ] [ pdf ] J. We introduce a modification of K-stability of a polarised variety which we conjecture to be equivalent to the existence of an extremal metric in the polarisation class.

The method also applies to analogous equations on compact Riemannian manifolds. This answers questions of Chen-Sun-Wang and He. A, 36 no.

Filtrations and test-configurations [ abstract ] [ pdf ] Math. We study the J-flow from the point of view of an algebro-geometric stability condition. We derive a priori estimates for solutions of a general szekeoyhidi of fully non-linear equations on compact Hermitian 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. This generalizes work of Arezzo-Pacard-Singer, who considered blowups in points. We show that the blowup of an extremal Kahler manifold at a relatively stable point in szekelyhdi sense of GIT admits an extremal metric in Kahler classes that make the exceptional divisor sufficiently small, extending a result of Arezzo-Pacard-Singer.

A variant for a complete extremal metric on the complement of a smooth divisor is also given. In terms of this we give a lower bound for the natural associated energy functional, and we show that the blowup behavior found by Fang-Lai is reflected by the optimal destabilizer.

We show that if a polarised manifold gaobr an extremal metric then it is K-polystable relative to a maximal torus of automorphisms.