In this paper, we prove a scalar curvature rigidity result for geodesic balls in Sn. This result contrasts sharply with the counterexamples to Min-Oo’s conjecture constructed in [7].

We introduce a gauge-theoretic integer valued lift of the Rohlin invariant of a smooth 4-manifold X with the homology of S1×S3. The invariant has two terms: one is a count of solutions to the Seiberg–Witten equations on X, and the other is essentially the index of the Dirac operator on a non-compact manifold with end modeled on the infinite cyclic cover of X. Each term is metric (and perturbation) dependent, and we show that these dependencies cancel as the metric and perturbation vary in a generic 1-parameter family.

We define analytic torsion (X, E,H) 2 detH•(X, E,H) for the twisted de Rham complex, consisting of the spaces of differential forms on a compact oriented Riemannian manifold X valued in a flat vector bundle E, with a differential given by rE + H ^ · , where rE is a flat connection on E, H is an odd-degree closed differential form on X, and H•(X, E,H) denotes the cohomology of this Z2-graded complex. The definition uses pseudodifferential operators and residue traces. We show that when dimX is odd, (X, E,H) is independent of the choice of metrics on X and E and of the representative H in the cohomology class [H]. We define twisted analytic torsion in the context of generalized geometry and show that when H is a 3-form, the deformation H 7! H−dB, where B is a 2-form on X, is equivalent to deforming a usual metric g to a generalized metric (g,B). We demonstrate some basic functorial properties. When H is a top-degree form, we compute the torsion, define its simplicial counterpart, and prove an analogue of the Cheeger-M¨uller Theorem. We also study the twisted analytic torsion for T -dual circle bundles with integral 3- form fluxes.

We show that for every simple closed curve , the extremal length and the hyperbolic length of  are quasi-convex functions along any Teichm¨uller geodesic. As a corollary, we conclude that, in Teichm¨uller space equipped with the Teichm¨uller metric, balls are quasi-convex.

Define a “Liouville domain” to be a compact exact symplectic manifold with contact-type boundary. We use embedded contact homology to assign to each four-dimensional Liouville domain (or subset thereof) a sequence of real numbers, which we call “ECH capacities”. The ECH capacities of a Liouville domain are defined in terms of the “ECH spectrum” of its boundary, which measures the amount of symplectic action needed to represent certain classes in embedded contact homology. Using cobordism maps on embedded contact homology (defined in joint work with Taubes), we show that the ECH capacities are monotone with respect to symplectic embeddings. We compute the ECH capacities of ellipsoids, polydisks, certain subsets of the cotangent bundle of T 2, and disjoint unions of examples for which the ECH capacities are known. The resulting symplectic embedding obstructions are sharp in some interesting cases, for example for the problem of embedding an ellipsoid into a ball (as shown by McDuff-Schlenk) or embedding a disjoint union of balls into a ball. We also state and present evidence for a conjecture under which the asymptotics of the ECH capacities of a Liouville domain recover its symplectic volume.