Let M 4n be a complete quaternionic K¨ahler manifold with scalar curvature bounded below by .16n(n + 2). We get a sharp estimate for the first eigenvalue λ1(M) of the Laplacian, which is λ1(M) ≤ (2n + 1)2. If the equality holds, then either M has only one end, or M is diffeomorphic to R × N with N given by a compact manifold. Moreover, if M is of bounded curvature, M is covered by the quaterionic hyperbolic space QHn and N is a compact quotient of the generalized Heisenberg group. When λ1(M) ≥ 8(n+2) 3 , we also prove that M must have only one end with infinite volume.

We study harmonic maps from an admissible flat simplicial complex to a non-positively curved Riemannian manifold. Our main regularity theorem is that these maps are C1, at the interfaces of the top-dimensional simplices in addition to satisfying a balancing condition. If we assume that the domain is a 2-complex,then these maps are C1. As an application, we show that the regularity, the balancing condition and a Bochner formula lead to rigidity and vanishing theorems for harmonic maps. Furthermore, we give an explicit relationship between our techniques and those obtained via combinatorial methods.

Given an acyclic representation  of the fundamental group of a compact oriented odd-dimensional manifold, which is close enough to an acyclic unitary representation, we define a refinement T of the Ray-Singer torsion associated to , which can be viewed as the analytic counterpart of the refined combinatorial torsion introduced by Turaev. T is equal to the graded determinant of the odd signature operator up to a correction term, the metric anomaly, needed to make it independent of the choice of the Riemannian metric.T is a holomorphic function on the space of such representations of the fundamental group. When  is a unitary representation, the absolute value of T is equal to the Ray-Singer torsion and the phase of T is proportional to the -invariant of the odd signature operator. The fact that the Ray-Singer torsion and the-invariant can be combined into one holomorphic function allows one to use methods of complex analysis to study both invariants. In particular, using these methods we compute the quotient of the refined analytic torsion and Turaev’s refinement of the combinatorial torsion generalizing in this way the classical Cheeger-M¨uller theorem. As an application, we extend and improve a result of Farber about the relationship between the Farber-Turaev absolute torsion and the -invariant. As part of our construction of T we prove several new results about determinants and -invariants of non self-adjoint elliptic operators.

Being inspired by a work of Curtis T. McMullen about a very impressive automorphism of a K3 surface of Picard number zero, we shall clarify the structure of the bimeromorphic automorphism group of a non-projective hyperk¨ahler manifold, up to finite group factor.

We construct noncompact solutions to the affine normal flow of hypersurfaces, and show that all ancient solutions must be either ellipsoids (shrinking solitons) or paraboloids (translating solitons). We also provide a new proof of the existence of a hyperbolic affine sphere asymptotic to the boundary of a convex cone containing no lines, which is originally due to Cheng-Yau. The main techniques are local second-derivative estimates for a parabolic Monge-Amp`ere equation modeled on those of Ben Andrews and Guti´errez-Huang, a decay estimate for the cubic form under the affine normal flow due to Ben Andrews, and a hypersurface barrier due to Calabi.