We discuss the concepts of energy and mass in relativity. On a finitely extended spatial region, they lead to the notion of quasilocal energy/mass for the boundary 2-surface in spacetime. A new definition was found in [27] that satisfies the positivity, rigidity, and asymptotics properties. The definition makes use of the surface Hamiltonian term which arises from Hamilton-Jacobi analysis of the gravitation action. The reference surface Hamiltonian is associated with an isometric embedding of the 2-surface into the Minkowski space. We discuss this new definition of mass as well as the reference surface Hamiltonian. Most of the discussion is based on joint work with PoNing Chen and Shing-Tung Yau.

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We study minimal hypersurfaces in manifolds of non-negative Ricci curvature, Euclidean volume growth and quadratic curvature decay at infinity. By comparison with capped spherical cones, we identify a precise borderline for the Ricci curvature decay. Above this value, no complete area-minimizing hypersurfaces exist. Below this value, in contrast, we construct examples.

In this paper, we study the behavior of Ricci-flat K¨ahler metrics on Calabi–Yau manifolds under algebraic geometric surgeries: extremal transitions or flops. We prove a version of Candelas and de la Ossa’s conjecture: Ricci-flat Calabi–Yau manifolds related by extremal transitions and flops can be connected by a path consisting of continuous families of Ricci-flat Calabi–Yau manifolds and a compact metric space in the Gromov–Hausdorff topology. In an essential step of the proof of our main result, the convergence of Ricci-flat K¨ahler metrics on Calabi–Yau manifolds along a smoothing is established, which can be of independent interest.

By using variational method and under generic condition, we show that Arnold diffusion exists in a priori hyperbolic and timeperiodic Hamiltonian systems with multiple degrees of freedom.