We study singular Hermitian metrics on vector bundles. There are two main
results in this paper. The first one is on the coherence of the higher rank analogue of multiplier
ideals for singular Hermitian metrics defined by global sections. As an application, we show
the coherence of the multiplier ideal of some positively curved singular Hermitian metrics whose
standard approximations are not Nakano semipositive. The aim of the second main result is to
determine all negatively curved singular Hermitian metrics on certain type of vector bundles, for
example, certain rank 2 bundles on elliptic curves.
Based on work of Rasmussen [Ras03], we construct a concordance invariant associated to the knot Floer complex, and exhibit examples in which this invariant gives arbitrarily better bounds on the 4-ball genus than the Ozsv ́ath-Szab ́o τ invariant.
Given an element in the first homology of a rational homology 3-sphere Y, one can consider the minimal rational genus of all knots in this homology class. This defines a function Θ on H1(Y;Z), which was introduced by Turaev as an analogue of Thurston norm. We will give a lower bound for this function using the correction terms in Heegaard Floer homology. As a corollary, we show that Floer simple knots in L-spaces are genus minimizers in their homology classes, hence answer questions of Turaev and Rasmussen about genus minimizers in lens spaces.
Two Dehn surgeries on a knot are called purely cosmetic, if they yield manifolds that are homeomorphic as oriented manifolds. Suppose there exist purely cosmetic surgeries on a knot in S^3, we show that the two surgery slopes must be the opposite of each other. One ingredient of our proof is a Dehn surgery formula for correction terms in Heegaard Floer homology.
We study the rigidity of polyhedral surfaces and the moduli
space of polyhedral surfaces using variational principles. Curvaturelike
quantities for polyhedral surfaces are introduced and are shown
to determine the polyhedral metric up to isometry. The action
functionals in the variational approaches are derived from the cosine
law. They can be considered as 2-dimensional counterparts of
the Schlaefli formula.