We consider minimal surfaces M which are complete, embedded, and have finite total curvature in R3, and bounded, entire solutions with finite Morse index of the Allen-Cahn equation
u+ f(u) = 0 in R3. Here f = −W0 with W bi-stable and balanced, for instance W(u) = 1 4 (1 − u2)2. We assume that M has m  2 ends, and additionally that M is non-degenerate, in the sense that its bounded Jacobi fields are all originated from rigid motions (this is known for instance for a Catenoid and for the Costa-Hoffman- Meeks surface of any genus). We prove that for any small  > 0, the Allen-Cahn equation has a family of bounded solutions depending on m − 1 parameters distinct from rigid motions, whose level sets are embedded surfaces lying close to the blown-up surface M := −1M, with ends possibly diverging logarithmically from M. We prove that these solutions are L1-non-degenerate up to rigid motions, and find that their Morse index coincides with the index of the minimal surface. Our construction suggests parallels of De Giorgi conjecture for general bounded solutions of finite Morse index.

Using the weak solution of Inverse mean curvature flow, we prove the sharp Minkowski-type inequality for outward minimizing hypersurfaces in Schwarzschild space.

We prove that a stable minimal hypersurface of an open ball which is immersed away from a closed (singular) set of finite codimension 2 Hausdorff measure and weakly close to a multiplicity 2 hyperplane must in the interior be the graph over the hyperplane of a 2-valued function satisfying a local C1, estimate. This regularity is optimal under our hypotheses. As a consequence, we also establish compactness of the class of stable minimal hypersur- faces of an open ball which have volume density ratios uniformly bounded by 3−δ for any fixed δ ∈ (0, 1) and interior singular sets of vanishing co-dimension 2 Hausdorff measure.

We prove the Poisson geometric version of the Local Reeb Stability (from foliation theory) and of the Slice Theorem (from equivariant geometry), which is also a generalization of Conn’s linearization theorem.

From any 4-dimensional oriented handlebody X without 3- and 4-handles and with b2  1, we construct arbitrary many compact Stein 4-manifolds that are mutually homeomorphic but not diffeomorphic to each other, so that their topological invariants (their fundamental groups, homology groups, boundary homology groups, and intersection forms) coincide with those of X. We also discuss the induced contact structures on their boundaries. Furthermore, for any smooth 4-manifold pair (Z, Y ) such that the complement Z − int Y is a handlebody without 3- and 4-handles and with b2  1, we construct arbitrary many exotic embeddings of a compact 4-manifold Y 0 into Z, such that Y 0 has the same topological invariants as Y .