We prove higher regularity properties of inverse mean curvature flow in Euclidean space: A sharp lower bound for the mean
curvature is derived for star-shaped surfaces, independently of the initial mean curvature. It is also shown that solutions to the inverse
mean curvature flow are smooth if the mean curvature is bounded from below. As a consequence we show that weak solutions
of the inverse mean curvature flow are smooth for large times,beginning from the first time where a surface in the evolution is
For a smooth projective toric surface we determine the Donaldson invariants and their wallcrossing in terms of the Nekrasov
partition function. Using the solution of the Nekrasov conjecture [33, 38, 3] and its refinement , we apply this result to give a
generating function for the wallcrossing of Donaldson invariants of good walls of simply connected projective surfaces with b+ = 1 in
terms of modular forms. This formula was proved earlier in  more generally for simply connected 4-manifolds with b+ = 1, as-
suming the Kotschick-Morgan conjecture, and it was also derived by physical arguments in .
The actions of a half Virasoro algebra have appeared in many integrable systems. In this paper we show that there is an action
of a (Half) Virasoro algebra on the space of (2+0) harmonic maps into a Lie group. This action is generated by a natural action
on the frames. A similar calculation on the space-time (1+1) harmonic maps yields formulas generated by John Schwarz.
Let X be a compact complex, not necessarily K¨ahler, manifold of dimension n. We characterize the volume of any holomorphic
line bundle L → X as the supremum R of the Monge-Amp`ere masses X Tn ac over all closed positive currents T in the first Chern class of L, where Tac is the absolutely continuous part of T in its Lebesgue decomposition. This result, new in the non-K¨ahler context, can be seen as holomorphic Morse inequalities for the cohomology of high tensor powers of line bundles endowed with arbitrarily singular Hermitian metrics. It gives, in particular, a new bigness criterion for line bundles in terms of existence of singular Hermitian metrics satisfying positivity conditions. The proof is based on the construction of a new regularization for closed (1, 1)-currents with a control of the Monge-Amp`ere masses of the approximating sequence. To this end, we prove a potential-theoretic result in one complex variable and study the growth of multiplier ideal sheaves associated with increasingly singular metrics.