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We solve the Plateau problem for marginally outer trapped surfaces in general Cauchy data sets. We employ the Perron method and tools from geometric measure theory to force and control a blow-up of Jang’s equation. Substantial new geometric insights regarding the lower order properties of marginally outer trapped surfaces are gained in the process. The techniques developed in this paper are flexible and can be adapted to other non-variational existence problems.

In this paper we study compact Sasaki manifolds in view of transverse K¨ahler geometry and extend some results in K¨ahler geometry to Sasaki manifolds. In particular we define integral invariants which obstruct the existence of transverse K¨ahler metric with harmonic Chern forms. The integral invariant f1 for the first Chern class case becomes an obstruction to the existence of transverse K¨ahler metric of constant scalar curvature. We prove the existence of transverse K¨ahler-Ricci solitons (or Sasaki-Ricci soliton) on compact toric Sasaki manifolds whose basic first Chern form of the normal bundle of the Reeb foliation is positive and the first Chern class of the contact bundle is trivial. We will further show that if S is a compact toric Sasaki manifold with the above assumption then by deforming the Reeb field we get a Sasaki-Einstein structure on S. As an application we obtain Sasaki-Einstein metrics on the U(1)-bundles associated with the canonical line bundles of toric Fano manifolds, including as a special case an irregular toric Sasaki-Einstein metrics on the unit circle bundle associated with the canonical bundle of the two-point blow-up of the complex projective plane.

We prove that the second positive Neumann eigenvalue of a bounded simply-connected planar domain of a given area does not exceed the first positive Neumann eigenvalue on a disk of half this area. The estimate is sharp and attained by a sequence of domains degenerating to a union of two identical disks. In particular, this result implies the P´olya conjecture for the second Neumann eigenvalue. The proof is based on a combination of analytic and topological arguments. As a by-product of our method we obtain an upper bound on the second eigenvalue for conformally round metrics on odd-dimensional spheres.

On a compact K¨ahler manifold we introduce a cohomological obstruction to the solvability of the constant scalar curvature (cscK) equation twisted by a semipositive form, appearing in works of Fine and Song-Tian. As a special case we find an obstruction for a manifold to be the base of a holomorphic submersion carrying a cscK metric in certain “adiabatic” classes. We apply this to find new examples of general type threefolds with classes which do not admit a cscK representative. When the twist vanishes our obstruction extends the slope stability of Ross-Thomas to effective divisors on a K¨ahler manifold. Thus we find examples of non-projective slope unstable manifolds.

We consider surfaces M2 immersed in En c × R, where En c is a simply connected n-dimensional complete Riemannian manifold with constant sectional curvature c 6= 0, and assume that the mean curvature vector of the immersion is parallel in the normal bundle. We consider further a Hopf-type complex quadratic form Q on M2, where the complex structure of M2 is compatible with the induced metric. It is not hard to check that Q is holomorphic (see [3], p.289). We will use this fact to give a reasonable description of immersed surfaces in En c ×R that have parallel mean curvature vector.

We study the question of local and global uniqueness of completions, based on null geodesics, of Lorentzian manifolds. We show local uniqueness of such boundary extensions. We give a necessary and sufficient condition for existence of unique maximal completions. The condition is verified in several situations of interest. This leads to existence and uniqueness of maximal spacelike conformal boundaries, of maximal strongly causal boundaries, as well as uniqueness of conformal boundary extensions for asymptotically simple space-times. Examples of applications include the definition of mass, or the classification of inequivalent extensions across a Cauchy horizon of the Taub space-time.

In this paper, we establish an analytic foundation for a fully non-linear equation 2 1 = f on manifolds with metrics of positive scalar curvature and apply it to give a (rough) classification of such manifolds. A crucial point is a simple observation that this equation is a degenerate elliptic equation without any condition on the sign of f and it is elliptic not only for f > 0 but also for f < 0. By defining a Yamabe constant Y2,1 with respect to this equation, we show that a manifold with metrics of positive scalar curvature admits a conformal metric of positive scalar curvature and positive 2-scalar curvature if and only if Y2,1 > 0. We give a complete solution for the corresponding Yamabe problem. Namely, let g0 be a positive scalar curvature metric, then in its conformal class there is a conformal metric with 2(g) = 1(g), for some constant . Using these analytic results, we give a rough classification of the space of manifolds with metrics of positive scalar curvature.

We study the spaces of locally finite stability conditions on the derived categories of coherent sheaves on the minimal resolutions of An-singularities supported at the exceptional sets. Our main theorem is that they are connected and simply-connected. The proof is based on the study of spherical objects in [30] and the homological mirror symmetry for An-singularities.

We construct many self-similar and translating solitons for Lagrangian mean curvature flow, including self-expanders and translating solitons with arbitrarily small oscillation on the Lagrangian angle. Our translating solitons play the same role as cigar solitons in Ricci flow, and are important in studying the regularity of Lagrangian mean curvature flow. Given two transverse Lagrangian planes Rn in Cn with sum of characteristic angles less than π, we show there exists a Lagrangian self-expander asymptotic to this pair of planes. The Maslov class of these self-expanders is zero. Thus they can serve as local models for surgeries on Lagrangian mean curvature flow. Families of self-shrinkers and self-expanders with different topologies are also constructed. This paper generalizes the work of Anciaux [1], Joyce [12], Lawlor [15], and Lee and Wang [18, 19].

Suppose L is a lamination of a Riemannian manifold by hypersurfaces with the same constant mean curvature H. We prove that every limit leaf of L is stable for the Jacobi operator. A simple but important consequence of this result is that the set of stable leaves of L has the structure of a lamination.