The existence of topologically slice knots that are of infinite order in the knot concordance group followed from Freedman’s work on topological surgery and Donaldson’s gauge theoretic approach to four-manifolds. Here, as an application of Ozsváth and Szabó’s Heegaard Floer theory, we show the existence of an infinite subgroup of the smooth concordance group generated by topologically slice knots of concordance order two. In addition, no nontrivial element in this subgroup can be represented by a knot with Alexander polynomial one.

We prove a lower bound for the number of negative eigenvalues for a Schrödinger operator on a Riemannian manifold via the integral of the potential.

We provide a rigorous perturbative quantization of the B-twisted topological sigma model via a first-order quantum field theory on derived mapping space in the formal neighborhood of constant maps. We prove that the first Chern class of the target manifold is the obstruction to the quantization via Batalin–Vilkovisky formalism. When the first Chern class vanishes, i.e. on Calabi–Yau manifolds, the factorization algebra of observables gives rise to the expected topological correlation functions in the B-model. We explain a twisting procedure to generalize to the Landau–Ginzburg case, and show that the resulting topological correlations coincide with Vafa’s residue formula.

Let $M$ be a complete Kähler manifold with nonnegative bisectional curvature. Suppose the universal cover does not split and $M$ admits a nonconstant holomorphic function with polynomial growth; we prove $M$ must be of maximal volume growth. This confirms a conjecture of Ni in “A monotonicity formula on complete Kähler manifolds with nonnegative bisectional curvature”, [J. Amer. Math. Soc. 17 (2004), 909–946, MR 2083471, Zbl 1071.58020]. There are two essential ingredients in the proof: the Cheeger–Colding theory on Gromov–Hausdorff convergence of manifolds, and the three-circle theorem for holomorphic functions in “Three circle theorems on Kähler manifolds and applications” by G. Liu [Arxiv: 1308.0710].

Consider a sequence of minimal varieties $M_i$ in a Riemannian manifold $N$ such that the measures of the boundaries are uniformly bounded on compact sets. Let $Z$ be the set of points at which the areas of the $M_i$ blow up. We prove that $Z$ behaves in some ways like a minimal variety without boundary. In particular, it satisfies the same maximum and barrier principles that a smooth minimal submanifold satisfies. For suitable open subsets $W$ of $N$, this allows one to show that if the areas of the $M_i$ are uniformly bounded on compact subsets of $W$, then the areas are in fact uniformly bounded on all compact subsets of $N$. Similar results are proved for varieties with bounded mean curvature. The results about area blow-up sets are used to show that the Allard Regularity Theorems can be applied in some situations where key hypotheses appear to be missing. In particular, we prove a version of the Allard Boundary Regularity Theorem that does not require any area bounds. For example, we prove that if a sequence of smooth minimal submanifolds converge as sets to a subset of a smooth, connected, properly embedded manifold with nonempty boundary, and if the convergence of the boundaries is smooth, then the convergence is smooth everywhere.