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.

The geometric Cauchy problem for a class of surfaces in a pseudo-Riemannian manifold of dimension 3 is to nd 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 noncharacteristic, and for characteristic initial curves (asymptotic curves for pseudospherical surfaces and null curves for timelike CMC) it is necessary and sucient for similar data to be prescribed along an additional characteristic curve that intersects the rst. The proofs also give a means of constructing all solutions using loop group techniques. The method used is the innite 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.

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 .

Some new differentiable sphere theorems are obtained via the Ricci flow and stable currents. We prove that if Mn is a compact manifold whose normalized scalar curvature and sectional curvature satisfy the pointwise pinching condition R0 > nKmax, where n 2 ( 1 4 , 1) is an explicit positive constant, then M is diffeomorphic to a spherical space form. We also provide a partial answer to Yau’s conjecture on the pinching theorem. Moreover, we prove that if Mn(n  3) is a compact manifold whose (n . 2)-th Ricci curvature and normalized scalar curvature satisfy the pointwise condition Ric(n.2) min > n(n .2)R0, where n 2 ( 1 4 , 1) is an explicit positive constant, then M is diffeomorphic to a spherical space form. We then extend the sphere theorems above to submanifolds in a Riemannian manifold. Finally we give a classification of submanifolds with weakly pinched curvatures, which improves the differentiable pinching theorems due to Andrews, Baker, and the authors.

The holonomy of the ambient metrics of Nurowski’s conformal structures associated to generic real-analytic 2-plane fields on oriented 5-manifolds is investigated. It is shown that the holonomy is always contained in the split real form G2 of the exceptional Lie group, and is equal to G2 for an open dense set of 2-plane fields given by explicit conditions. In particular, this gives an infinitedimensional family of metrics of holonomy equal to split G2. These results generalize work of Leistner-Nurowski. The inclusion of the holonomy in G2 is established by proving an ambient extension theorem for parallel tractors in the context of conformal geometry in general signature and dimension, which is expected to be of independent interest.