Volume 2017, Issue 724

In this article we generalise the Witten deformation for stratified spaces and a class of Morse functions which we call radial Morse functions. In the first part of the article we perform the Witten deformation on the complex of L2${{\mathrm{L}}^{2}}$-forms on a (general) stratified space. A local Spectral Gap Theorem for the model Witten Laplacian near critical points of the Morse function is proved for general stratified spaces. The global Spectral Gap Theorem for the Witten Laplacian can be generalised to spaces with isolated singularities as well as to Witt spaces. In the second part of the article we give, in the case of isolated singularities, a geometric interpretation of the complex of eigenforms of the Witten Laplacian to small eigenvalues. For this, we have to establish first an analogue of the Morse–Thom–Smale complex.

We consider an inverse problem associated with some 2-dimensional non-compact surfaces with conical singularities, cusps and regular ends. Our motivating example is a Riemann surface ℳ=Γ\ℍ2${\mathcal{M}=\Gamma\backslash\mathbb{H}^{2}}$ associated with a Fuchsian group of the first kind Γ containing parabolic elements. The surface ℳ${\mathcal{M}}$ is then non-compact, and has a finite number of cusps and elliptic singular points, which is regarded as a hyperbolic orbifold. We introduce a class of Riemannian surfaces with conical singularities on its finite part, having cusps and regular ends at infinity, whose metric is asymptotically hyperbolic. By observing solutions of the Helmholtz equation at the cusp, we define a generalized S-matrix. We then show that this generalized S-matrix determines the Riemannian metric and the structure of conical singularities.

We prove several results on the lifespan, regularity, and uniqueness of solutions of the Cauchy problem for the homogeneous complex and real Monge–Ampère equations (HCMA/HRMA) under various a priori regularity conditions. We use methods of characteristics in both the real and complex settings to bound the lifespan of solutions with prescribed regularity. In the complex domain, we characterize the C3${C^{3}}$ lifespan of the HCMA in terms of analytic continuation of Hamiltonian mechanics and intersection of complex time characteristics. We use a conservation law type argument to prove uniqueness of solutions of the Cauchy problem for the HCMA. We then prove that the Cauchy problem is ill-posed in C3${C^{3}}$, in the sense that there exists a dense set of C3${C^{3}}$ Cauchy data for which there exists no C3${C^{3}}$ solution even for a short time. In the real domain we show that the HRMA is equivalent to a Hamilton–Jacobi equation, and use the equivalence to prove that any differentiable weak solution is smooth, so that the differentiable lifespan equals the convex lifespan determined in our previous articles. We further show that the only obstruction to C1${C^{1}}$ solvability is the invertibility of the associated Moser maps. Thus, a smooth solution of the Cauchy problem for HRMA exists for a positive but generally finite time and cannot be continued even as a weak C1${C^{1}}$ solution afterwards. Finally, we introduce the notion of a “leafwise subsolution” for the HCMA that generalizes that of a solution, and many of our aforementioned results are proved for this more general object.

We study the problem of conformal deformation of Riemannian structure to constant scalar curvature with zero mean curvature on the boundary. We prove compactness for the full set of solutions when the boundary is umbilic and the dimension n≤24${n\leq 24}$. The Weyl Vanishing Theorem is also established under these hypotheses, and we provide counter-examples to compactness when n≥25${n\geq 25}$. Lastly, our methods point towards a vanishing theorem for the umbilicity tensor, which will be fundamental for a study of the non-umbilic case.

Let X be a compact Kähler manifold such that the anticanonical bundle -KX$-K_{X}$ is nef. A classical conjecture claims that the Albanese map X→T$X\to T$ is submersive. We prove this conjecture if the general fibre is a weak Fano manifold. If X is projective we prove the conjecture also for fibres of dimension at most two.