We find bounds for Weil-Petersson holomorphic sectional curvature, and the Weil-Petersson curvature operator in several regimes, that do not depend on the topology of the underlying surface. Among other results, we show that the minimal Weil-Petersson holomorphic sectional curvature of a sufficiently thick hyperbolic surface is comparable to $-1$, independently of the genus. This provides a counterexample to some suggestions that the Weil-Petersson metric becomes asymptotically flat, as the genus $g$ goes to infinity, in the thick loci in \tec space. Adopting a different perspective on curvature, we also show that the minimal (most negative) eigenvalue of the curvature operator at any point in the \tec space $\Teich(S_g)$ of a closed surface $S_g$ of genus $g$ is uniformly bounded away from zero. Restricting to a thick part of $\Teich(S_g)$, we show that the minimal eigenvalue is uniformly bounded below by an explicit constant which does not depend on the topology of the surface but only on the given bound on injectivity radius.

In this note, we study submanifold geometry of the Atiyah--Hitchin manifold, a double cover of the 2-monopole moduli space, which plays an important role in various settings such as the supersymmetric background of string theory. When the manifold is naturally identified as the total space of a line bundle over S^2, the zero section is a distinguished minimal 2-sphere of considerable interest. In particular, there has been a conjecture about the uniqueness of this minimal 2-sphere among all closed minimal 2-surfaces. We show that this minimal 2-sphere satisfies the ``strong stability condition" proposed in our earlier work, and confirm the global uniqueness as a corollary.

By studying the Seiberg-Witten equations on end-periodic manifolds, we give an obstruction on the existence of positive scalar curvature metric on compact 4-manifolds with the same homology as S1×S3. This obstruction is given in terms of the relation between the Frøyshov invariant of the generator of H3(X;Z) with the 4-dimensional Casson invariant λSW(X) defined by Mrowka-Ruberman-Saveliev. Along the way, we develop a framework that can be useful in further study of the Seiberg-Witten theory on general end-periodic manifolds.

In this paper, we solve a Monge-Ampre-type equation by the continuity method. This equation is motivated by the superstring theory.

In this paper we investigate the one-dimensional hyperbolic mean curvature flow for closed plane curves. More precisely, we consider a family of closed curves S 1[0, T) 2 which satisfies the following evolution equation 2 F t 2 (u, t)= k (u, t) N(u, t)- (u, t),(u, t) S 1[0, T) with the initial data F (u, 0)= F 0 (u) a n d F t (u, 0)= f (u) N 0, where k is the mean curvature and is the unit inner normal vector of the plane curve F (u, t), f (u) and are the initial velocity and the unit inner normal vector of the initial convex closed curve Fo, respectively, and is given by ( 2 F s t, F t) T, in which stands for the unit tangent vector. The above problem is an initial value problem for a system of partial differential equations for F, it can be completely reduced to an initial value problem for a single partial differential equation for its support function. The latter equation is a hyperbolic Monge-Ampere