Stable complete minimal surfaces in hyperkähler manifolds
AbstractIn this paper we prove that an isometric stable minimal immersion of a complete oriented surface into a hyperkähler 4-manifold is holomorphic with respect to an orthogonal complex structure, if...
View ArticleMoment Maps, Scalar Curvature and Quantization of Kähler Manifolds
AbstractBuilding on Donaldson’s work on constant scalar curvature metrics, we study the space of regular Kähler metrics Eω, i.e. those for which deformation quantization has been defined by Cahen, Gutt...
View ArticleBlowing up and desingularizing constant scalar curvature Kähler manifolds
AbstractThis paper is concerned with the existence of constant scalar curvature Kähler metrics on blow-ups at finitely many points of compact manifolds which already carry constant scalar curvature...
View ArticleStable bundles and the first eigenvalue of the Laplacian
AbstractIn this article we study the first eigenvalue of the Laplacian on a compact manifold using stable bundles and balanced bases. Our main result is the following: Let M be a compact Kähler...
View ArticleOn homothetic balanced metrics
AbstractIn this article, we study the set of balanced metrics given in Donaldson’s terminology (J. Diff. Geometry 59:479–522, 2001) on a compact complex manifold M which are homothetic to a given...
View ArticleSelf-shrinkers for the mean curvature flow in arbitrary codimension
AbstractIn this paper, we generalize Colding–Minicozzi’s recent results about codimension-1 self-shrinkers for the mean curvature flow to higher codimension. In particular, we prove that the sphere...
View ArticleSzegö kernel, regular quantizations and spherical CR-structures
AbstractWe compute the Szegö kernel of the unit circle bundle of a negative line bundle dual to a regular quantum line bundle over a compact Kähler manifold. As a corollary we provide an infinite...
View ArticleGeometric Constructions of Extremal Metrics on Complex Manifolds
AbstractIn this note we review recent progresses on the existence problem of Kähler constant scalar curvature metrics on complex manifolds. The content of this note is an expanded version of author’s...
View ArticleA variational characterization of complex submanifolds
AbstractIn this note, we generalize our results in Arezzo and Sun (Reine Angew Math, doi:10.1515/crelle-2013-0097, 2012) to integer p-currents of any degree. We prove that if the mass of a current, as...
View ArticleSome remarks on the symplectic and Kähler geometry of toric varieties
AbstractLet M be a projective toric manifold. We prove two results concerning, respectively, Kähler–Einstein submanifolds of M and symplectic embeddings of the standard Euclidean ball in M. Both...
View ArticleGeneralized Connected Sum Constructions for Resolutions of Extremal and Kcsc...
AbstractIn this note we review recent progresses on the existence problem of extremal and Kähler constant scalar curvature metrics on complex manifolds. The content of this note is an expanded version...
View ArticleOn the Curvature of Conic Kähler–Einstein Metrics
AbstractWe prove a regularity result for Monge–Ampère equations degenerate along smooth divisor on Kähler manifolds in Donaldson’s spaces of \(\beta \)-weighted functions. We apply this result to study...
View ArticleOn the Kummer construction for Kcsc metrics
AbstractGiven a compact constant scalar curvature Kähler orbifold, with nontrivial holomorphic vector fields, whose singularities admit a local ALE Kähler Ricci-flat resolution, we find sufficient...
View ArticleBig and Nef Classes, Futaki Invariant and Resolutions of Cubic Threefolds
AbstractIn this note we revisit and extend few classical and recent results on the definition and use of the Futaki invariant in connection with the existence problem for Kähler constant scalar...
View ArticleLocalization in the Discrete Non-linear Schrödinger Equation and Geometric...
AbstractIt is well known that, if the initial conditions have sufficiently high energy density, the dynamics of the classical Discrete Non-Linear Schrödinger Equation (DNLSE) on a lattice shows a form...
View Article