We further develop the asymptotic analytic approach to the study of scattering diagrams. We do so by analyzing the asymptotic behavior of Maurer--Cartan elements of a differential graded Lie algebra constructed from a (not-necessarily tropical) monoid-graded Lie algebra. In this framework, we give alternative differential geometric proofs of the consistent completion of scattering diagrams, originally proved by Kontsevich--Soibelman, Gross--Siebert and Bridgeland. We also give a geometric interpretation of theta functions and their wall-crossing. In the tropical setting, we interpret Maurer--Cartan elements, and therefore consistent scattering diagrams, in terms of the refined counting of tropical disks. We also describe theta functions, in both their tropical and Hall algebraic settings, in terms of distinguished flat sections of the Maurer--Cartan-deformed differential. In particular, this allows us to give a combinatorial description of Hall algebra theta functions for acyclic quivers with non-degenerate skew-symmetrized Euler forms.

In this paper, we establish a Bochner-type formula on Alexandrov spaces with Ricci curvature bounded below. Yau’s gradient estimate for harmonic functions is also obtained on Alexandrov spaces.

We prove flatness of complete Riemannian planes and cylinders without conjugate points under optimal conditions on the area growth.

Let X be a compact connected strongly pseudoconvex CR Manifold of real dimension 2n.1 in Cn+1. Tanaka introduced a spec- tral sequence E(p,q) r (X) with E(p,q) 1 (X) being the Kohn-Rossi cohomology group and E(k,0) 2 (X) being the holomorphic De Rham cohomology denoted by Hk h(X). We study the holomorphic De Rham cohomology in terms of the s-invariant of the isolated sin- gularities of the variety V bounded by X. We give a characterization of the singularities with vanishing s-invariants. For n ≥ 3, Yau used the Kohn-Rossi cohomology groups to solve the classical complex Plateau problem in 1981. For n = 2, the problem has re- mained unsolved for over a quarter of a century. In this paper, we make progress in this direction by putting some conditions on X so that V will have very mild singularities. Specifically, we prove that if dimX = 3 and H2 h(X) = 0, then X is a boundary of complex variety V with only isolated quasi-homogeneous singularities such that the dual graphs of the exceptional sets in the resolution are star shaped and all curves are rational.

Quasilocal mass in general relativity is a notion defined for a closed space like 2-surface in spacetime. In this note, we explain the definition in [23] and [24] from a mathematical viewpoint, emphasizing the connection to differential geometry and nonlinear partial differential equations. We also discuss a minimax interpretation of the definition and compare with other notions of quasilocal mass.