Global bifurcation for monotone fronts of elliptic equations

In this paper, we present two results on global continuation of monotone front-type solutions to elliptic PDEs posed on infinite cylinders. This is done under quite general assumptions, and in particular applies even to fully nonlinear equations as well as quasilinear problems with transmission boundary conditions. Our approach is rooted in the analytic global bifurcation theory of Dancer and Buffoni--Toland, but extending it to unbounded domains requires contending with new potential limiting behavior relating to loss of compactness. We obtain an exhaustive set of alternatives for the global behavior of the solution curve that is sharp, with each possibility having a direct analogue in the bifurcation theory of second-order ODEs. As a major application of the general theory, we construct global families of internal hydrodynamic bores. These are traveling front solutions of the full two-phase Euler equation in two dimensions. The fluids are confined to a channel that is bounded above and below by rigid walls, with incompressible and irrotational flow in each layer. Small-amplitude fronts for this system have been obtained by several authors. We give the first large-amplitude result in the form of continuous curves of elevation and depression bores. Following the elevation curve to its extreme, we find waves whose interfaces either overturn (develop a vertical tangent) or become exceptionally singular in that the flow in both layers degenerates at a single point on the boundary. For the curve of depression waves, we prove that either the interface overturns or it comes into contact with the upper wall.Comment: 60 pages, 6 figure

The effective Hamiltonian in curved quantum waveguides under mild regularity assumptions

The Dirichlet Laplacian in a curved three-dimensional tube built along a spatial (bounded or unbounded) curve is investigated in the limit when the uniform cross-section of the tube diminishes. Both deformations due to bending and twisting of the tube are considered. We show that the Laplacian converges in a norm-resolvent sense to the well known one-dimensional Schroedinger operator whose potential is expressed in terms of the curvature of the reference curve, the twisting angle and a constant measuring the asymmetry of the cross-section. Contrary to previous results, we allow the reference curves to have non-continuous and possibly vanishing curvature. For such curves, the distinguished Frenet frame standardly used to define the tube need not exist and, moreover, the known approaches to prove the result for unbounded tubes do not work. Our main ideas how to establish the norm-resolvent convergence under the minimal regularity assumptions are to use an alternative frame defined by a parallel transport along the curve and a refined smoothing of the curvature via the Steklov approximation.Comment: 29 pages, 6 figure

Spectral and scattering theory for symbolic potentials of order zero

The spectral and scattering theory is investigated for a generalization, to scattering metrics on two-dimensional compact manifolds with boundary, of the class of smooth potentials on the Euclidean plane which are homogeneous of degree zero near infinity. The most complete results require the additional assumption that the restriction of the potential to the circle(s) at infinity be Morse. Generalized eigenfunctions associated to the essential spectrum at non-critical energies are shown to originate both at minima and maxima, although the latter are not germane to the $L^2$ spectral theory. Asymptotic completeness is shown, both in the traditional $L^2$ sense and in the sense of tempered distributions. This leads to a definition of the scattering matrix, the structure of which will be described in a future publication.Comment: 69 page

Analytic torsion for surfaces with cusps II. Regularity, asymptotics and curvature theorem

In this article we study the Quillen norm on the determinant line bundle associated with a family of complex curves with cusps, which admit singular fibers. More precisely, we fix a family of complex curves $\pi : X \to S$, which admit at most double-point singularities. Let $(\xi, h^{\xi})$ be a holomorphic Hermitian vector bundle over $X$. Let $\sigma_1, \ldots, \sigma_m : S \to X$ be disjoint holomorphic sections. We denote the divisor $D_{X/S} := \rm{Im}(\sigma_1) + \cdots + \rm{Im}(\sigma_m)$, and endow the relative canonical line bundle $\omega_{X/S}$ with a Hermitian norm such that its restriction at each fiber of $\pi$ induces K\"ahler metric with hyperbolic cusps. This Hermitian norm induces the Hermitian norm on the twisted relative canonical line bundle $\omega_{X/S}(D) := \omega_{X/S} \otimes \mathscr{O}_{X}(D_{X/S})$. For $n \leq 0$, we study the determinant line bundle $\lambda(j^*(\xi \otimes \omega_{X/S}(D)^n))$. We endow it with the Quillen norm by using the analytic torsion from the first paper of this series. Then we study the regularity of this Quillen norm and its asymptotics near the locus of singular curves. The singular terms of the asymptotics turn out to be reasonable enough, so that the curvature of $\lambda(j^*(\xi \otimes \omega_{X/S}(D)^n))$ is well-defined as a current over $S$. We derive the explicit formula for this current, which gives a refinement of Riemann-Roch-Grothendieck theorem at the level of currents. This generalizes the curvature formulas of Takhtajan-Zograf and Bismut-Bost. As a consequence of our study, we also get some regularity results on the Weil-Petersson form over the moduli space of pointed curves, which are enough to conclude the well-known fact, originally due to Wolpert, that the Weil-Petersson volume of the moduli space of pointed stable curves is a rational multiple of a power of $\pi$.Comment: 38 pages, 1 figur

Smooth perturbations of the functional calculus and applications to Riemannian geometry on spaces of metrics

We show for a certain class of operators $A$ and holomorphic functions $f$ that the functional calculus $A\mapsto f(A)$ is holomorphic. Using this result we are able to prove that fractional Laplacians $(1+\Delta^g)^p$ depend real analytically on the metric $g$ in suitable Sobolev topologies. As an application we obtain local well-posedness of the geodesic equation for fractional Sobolev metrics on the space of all Riemannian metrics.Comment: 31 page