We embed polarised orbifolds with cyclic stabiliser groups into weighted projective space via a weighted form of Kodaira embedding. Dividing by the (non-reductive) automorphisms of weighted projective space then formally gives a moduli space of orbifolds. We show how to express this as a reductive quotient and so a GIT problem, thus defining a notion of stability for orbifolds. We then prove an orbifold version of Donaldson’s theorem: the existence of an orbifold K¨ahler metric of constant scalar curvature implies K-semistability. By extending the notion of slope stability to orbifolds, we therefore get an explicit obstruction to the existence of constant scalar curvature orbifold K¨ahler metrics. We describe the manifold applications of this orbifold result, and show how many previously known results (Troyanov, Ghigi-Koll´ar, Rollin-Singer, the AdSCFT Sasaki-Einstein obstructions of Gauntlett-Martelli-Sparks-Yau) fit into this framework.

This chapter discusses some recent results by Richard Schoen and Shing-Tung Yau on the structure of manifolds with positive scalar curvature. The chapter presents theorems which are felt to provide a more complete picture of manifolds with positive scalar curvature: (1) let M be a compact four-dimensional manifold with positive scalar curvature. Then there exists no continuous map with non-zero degree onto a compact K(,1). (2) Let M be n-dimensional complete manifold with non-negative scalar curvature. Then any conformed immersion of M into Sⁿ is one to one. In particular, any complete conformally flat manifold with non-negative scalar curvature is the quotient of a domain in Sⁿ by a discrete subgroup of the conformal group. (3.) Let M be a compact manifold whose fundamental group is not of exponential growth. Then unless M is covered by Sⁿ, Sⁿ¹ x S¹ or the torus, M admits no

We develop a Floer theoretical gluing technique and apply it to deal with the most generic singular ber in the SYZ program, namely the product of a torus with the immersed two-sphere with a single nodal self-intersection. As an application, we construct immersed Lagrangians in Gr(2;Cn) and OG(1;C5) and derive their SYZ mirrors. It recovers the Lie theoretical mirrors constructed by Rietsch. It also gives an eective way to compute stable disks (with non-trivial obstructions) bounded by immersed Lagrangians.

In this survey, we discuss some recent results on free boundary minimal surfaces in the Euclidean unit-ball. The subject has been a very active field of research in the past few years due to the seminal work of Fraser and Schoen on the extremal Steklov eigenvalue problem. We review several different techniques of constructing examples of embedded free boundary minimal surfaces in the unit ball. Next, we discuss some uniqueness results for free boundary minimal disks and the conjecture about the uniqueness of critical catenoid. We also discuss several Morse index estimates for free boundary minimal surfaces. Moreover, we describe estimates for the first Steklov eigenvalue on such free boundary minimal surfaces and various smooth compactness results. Finally, we mention some sharp area bounds for free boundary minimal submanifolds and related questions.

IT IS well-known that if a compact group acts differentiably on a differentiable manifold, then this group must preserve some Riemannian metric on this manifold. From this point of view, we shall discuss certain facts about group actions on manifolds. In 01, we study the group of isometries of a non-compact manifold. We discover that if the group is non-compact, the manifold has to split as a product of the Euclidean space and another manifold. This gives some information on problem 33 of [5]. It also gives some information on the group of biholomorphic transformations of a complex manifold whose Bergman metric is not trivial. Then we find a topological obstruction for a non-compact manifold to admit an infinite group to act freely and properly discontinuously. Namely, we prove that the natural map from the de Rham cohomology group with compact support to the de Rham cohomology group without compact support is trivial when the first group has finite dimension. In $2, we obtain some topological obstructions for group actions by looking at the complex of invariant differential forms. We prove, for example, that if a compact group acts on a compact manifold with non-zero Euler number, then wI A*** A ut+,= 0 for all closed invariant l-forms wl,..., Ok+ 1 with k 2 the codimension of the principle orbit.(It may be interesting to note that the vanishing is on the form level so that any secondary obstruction should also vanish.) We also prove a topological version of this theorem for circle actions. An interesting corollary is that if a manifold is the connected sum of a torus and a compact manifold with Euler number# 2, then it does not admit any circle actions