We consider the zeros on the boundary @­ of a Neumann eigen- function '¸j of a real analytic plane domain ­. We prove that the number of its boundary zeros is O(¸j) where ¡¢'¸j = ¸2j '¸j . We also prove that the number of boundary critical points of either a Neumann or Dirichlet eigenfunction is O(¸j ). It follows that the number of nodal lines of '¸j (components of the nodal set) which touch the boundary is of order ¸j . This upper bound is of the same order of magnitude as the length of the total nodal line, but is the square root of the Courant bound on the number of nodal components in the interior. More generally, the results are proved for piecewise analytic domains.

Given a Riemannian manifold and a closed submanifold, we find a geodesic segment with free boundary on the given submanifold. This is a corollary of the min–max theory which we develop in this article for the free boundary variational problem. In particular, we develop a modified Birkhoff curve shortening process to achieve a strong “Colding–Minicozzi” type min–max approximation result.

We consider inverse curvature ows in Hn+1 with star-shaped initial hypersurfaces and prove that the ows exist for all time, and that the leaves converge to innity, become strongly convex exponentially fast and also more and more totally umbilic. Afteran appropriate rescaling the leaves converge in C1 to a sphere.

We state and prove a Chern–Osserman-type inequality in terms of the volume growth for minimal surfaces$S$which have finite total extrinsic curvature and are properly immersed in a Cartan–Hadamard manifold$N$with sectional curvatures bounded from above by a negative quantity$K$_{$N$}≤$b$<0 and such that they are not too curved (on average) with respect to the hyperbolic space with constant sectional curvature given by the upper bound$b$. We also prove the same Chern–Osserman-type inequality for minimal surfaces with finite total extrinsic curvature and properly immersed in an asymptotically hyperbolic Cartan–Hadamard manifold$N$with sectional curvatures bounded from above by a negative quantity$K$_{$N$}≤$b$<0.

This article studies the mean curvature flow of Lagrangian submanifolds. In particular, we prove the following global existence and convergence theorem: if the potential function of a Lagrangian graph in $T^{2n}$ is convex, then the flow exists for all time and converges smoothly to a flat Lagrangian submanifold. We also discuss various conditions on the potential function that guarantee global existence and convergence.