Let $(X,D)$ be a logarithmic pair, and let $h$ be a smooth metric on $T_{X \setminus D}$. We give a simple criterion on the curvature of $h$ for the bigness of $\Omega_X(\operatorname{log} D)$ or $\Omega_X$. As an application, we obtain a metric proof of the bigness of $\Omega_X(\operatorname{log} D)$ on any toroidal compactification of a bounded symmetric domain. Then, we use this singular metric approach to study the bigness and the nefness of $\Omega_X$ in the more specific case of the ball. We obtain effective ramification orders for a cover $X^\prime \to X$, étale outside the boundary, to have all its subvarieties with big cotangent bundle. We also prove that $\Omega_{X^\prime}$ is nef if the ramification is high enough. Moreover, the ramification orders we obtain do not depend on the dimension of the ball quotient we consider.

The wedge product on de Rham complex of a Riemannian manifold $M$ can be pulled back to $H^\ast (M)$ via explicit homotopy constructed by using Green’s operator which gives higher product structures. We prove Fukaya’s conjecture which suggests that Witten deformation of these higher product structures has semiclassical limits as operators defined by counting gradient flow trees with respect to Morse functions, which generalizes the remarkable Witten deformation of de Rham differential from a statement concerning homology to one concerning real homotopy type of $M$. Various applications of this conjecture to mirror symmetry are also suggested by Fukaya in [6].

Let $E_{\tau}:= \mathbb{C}/(\mathbb{Z}+ \mathbb{Z} \tau)$ be a flat torus and $G(z; \tau)$ be the Green function on $E_{\tau}$. Consider the multiple Green function $G_{n}$ on$(E_{\tau})^{n}$: \[ G_n (z_1, \cdots ,z_n ; \tau) := \sum_{i \lt j} G(z_i - z_j ; \tau) - n \sum_{i=1}^n G(z_i ; \tau). \] We prove that for $ \tau \in i \mathbb{R}_{\gt 0}$, i.e. $E_\tau$ is a rectangular torus, $G_n$ has exactly $2n + 1$ critical points modulo the permutation group $S_n$ and all critical points are non-degenerate. More precisely, there are exactly $n$ (resp. $n+1$) critical points $ \boldsymbol{a}$’s with the Hessian satisfying $(-1)^n \det D^2 G_n (\boldsymbol{a} ; \tau) \lt 0$ (resp. $\gt 0$). This confirms a conjecture in [4]. Our proof is based on the connection between $G_n$ and the classical Lamé equation from [4, 19], and one key step is to establish a precise formula of the Hessian of critical points of $G_{n}$ in terms of the monodromy data of the Lamé equation. As an application, we show that the mean field equation \[ \Delta_u + e^u = \rho \delta_0 \textrm{ on } E_\tau \] has exactly $n$ solutions for $8 \pi n - \rho \gt 0$ small, and exactly $n+1$ solutions for $\rho - 8 \pi n \gt 0$ small.

In the 1960s, Almgren [3, 4] initiated a program to find minimal hypersurfaces in Riemannian manifolds using min-max method. This program was largely advanced by Pitts [34] and Schoen–Simon [37] in the 1980s when the manifold is compact without boundary. In this paper, we finish this program for general compact manifold with non-empty boundary. As a corollary, we establish the existence of a smooth embedded minimal hypersurface with non-empty free boundary in any compact smooth Euclidean domain. An application of our general existence result combined with the work of Marques and Neves [31] shows that for any compact Riemannian manifolds with nonnegative Ricci curvature and convex boundary, there exist infinitely many properly embedded minimal hypersurfaces with non-empty free boundary.

We show that for every quasi-isometric map from a Hadamard manifold of pinched negative curvature to a proper, Gromov hyperbolic, $\operatorname{CAT}(0)$-space there exists an energy minimizing harmonic map at finite distance. This harmonic map is moreover Lipschitz. This generalizes a recent result of Benoist–Hulin.