From any 4-dimensional oriented handlebody $X$ without 3- and 4-handles and with $b_2 \ge 1$, 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 $Z − \operatorname{int} Y$ is a handlebody without 3- and 4-handles and with $b_2 \ge 1$, we construct arbitrary many exotic embeddings of a compact 4-manifold $Y'$ into $Z$, such that $Y'$ has the same topological invariants as $Y$.

The geometric Cauchy problem for a class of surfaces in a pseudo-Riemannian manifold of dimension 3 is to find the surface which contains a given curve with a prescribed tangent bundle along the curve. We consider this problem for constant negative Gauss curvature surfaces (pseudospherical surfaces) in Euclidean 3-space, and for timelike constant non-zero mean curvature (CMC) surfaces in the Lorentz-Minkowski 3-space. We prove that there is a unique solution if the prescribed curve is non-characteristic, and for characteristic initial curves (asymptotic curves for pseudospherical surfaces and null curves for timelike CMC) it is necessary and suffcient for similar data to be prescribed along an additional characteristic curve that intersects the first. The proofs also give a means of constructing all solutions using loop group techniques. The method used is the infinite dimensional d'Alembert type representation for surfaces associated with Lorentzian harmonic maps (1-1 wave maps) into symmetric spaces, developed since the 1990's. Explicit formulae for the potentials in terms of the prescribed data are given, and some applications are considered.

We consider minimal surfaces $M$ which are complete, embedded, and have finite total curvature in $\mathbb{R}^3$, and bounded, entire solutions with finite Morse index of the Allen-Cahn equation $\Delta u+ f(u) = 0$ in $\mathbb{R}^3$. Here $f = −W'$ with $W$ bi-stable and balanced, for instance $W(u) = \frac{1}{4} (1 − u^2)^2$. We assume that $M$ has $m \ge 2$ 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 $\alpha \gt 0$, 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 $M_\alpha := \alpha^{−1}M$, with ends possibly diverging logarithmically from $M_\alpha$. We prove that these solutions are $L^\infty$-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.

A Riemannian or Finsler metric on a compact manifold $M$ gives rise to a length function on the free loop space $\Lambda M$, whose critical points are the closed geodesics in the given metric. If $X$ is a homology class on $\Lambda M$, the “minimax” critical level $\mathsf{cr}(X)$ is a critical value. Let $M$ be a sphere of dimension $\gt 2$, and fix a metric $g$ and a coefficient field $G$. We prove that the limit as $\deg(X)$ goes to infinity of $\mathsf{cr}(X)/ \deg(X)$ exists. We call this limit $\overline\alpha = \overline\alpha(M, g,G)$ the global mean frequency of $M$. As a consequence we derive resonance statements for closed geodesics on spheres; in particular either all homology on $\Lambda$ of sufficiently high degreee lies hanging on closed geodesics of mean frequency (length/average index) $\overline\alpha$, or there is a sequence of infinitely many closed geodesics whose mean frequencies converge to $\overline\alpha$. The proof uses the Chas-Sullivan product and results of Goresky-Hingston.