The authors study noncompact surfaces evolving by mean curvature flow (mcf). For an open set of initial data that are $C^3$-close to round, but without assuming

Ricci flow is a powerful technique using a heat-type equation to deform Riemannian metrics on manifolds to better metrics in the search for geometric decomposit

In a previous study, the authors built the Bellman function for integral functionals on the space. The present paper provides a development of the subject. They

The authors consider the full irrotational water waves system with surface tension and no gravity in dimension two (the capillary waves system), and prove globa

For any cluster algebra whose underlying combinatorial data can be encoded by a bordered surface with marked points, the authors construct a geometric realizati