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Register a new userThe diffractive wave trace on manifolds with conic singularities
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Let $(X,g)$ be a compact manifold with conic singularities. Taking $Delta_g$ to be the Friedrichs extension of the Laplace-Beltrami operator, we examine the singularities of the trace of the half-wave group $e^{- i t sqrt{ smash[b]{Delta_g}}}$ arising from strictly diffractive closed geodesics. Under a generic nonconjugacy assumption, we compute the principal amplitude of these singularities in terms of invariants associated to the geodesic and data from the cone point. This generalizes the classical theorem of Duistermaat-Guillemin on smooth manifolds and a theorem of Hillairet on flat surfaces with cone points.
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We revisit the paper [Mel86] by R. Melrose, providing a full proof of the main theorem on propagation of singularities for subelliptic wave equations, and linking this result with sub-Riemannian geometry. This result asserts that singularities of subelliptic wave equations only propagate along null-bicharacteristics and abnormal extremal lifts of singular curve. As a new consequence, for x = y and denoting by K G the wave kernel, we obtain that the singular support of the distribution t $rightarrow$ K G (t, x, y) is included in the set of lengths of the normal geodesics joining x and y, at least up to the time equal to the minimal length of a singular curve joining x and y.
We consider manifolds with conic singularites that are isometric to $mathbb{R}^{n}$ outside a compact set. Under natural geometric assumptions on the cone points, we prove the existence of a logarithmic resonance-free region for the cut-off resolvent. The estimate also applies to the exterior domains of non-trapping polygons via a doubling process. The proof of the resolvent estimate relies on the propagation of singularities theorems of Melrose and the second author to establish a very weak Huygens principle, which may be of independent interest. As applications of the estimate, we obtain a exponential local energy decay and a resonance wave expansion in odd dimensions, as well as a lossless local smoothing estimate for the Schr{o}dinger equation.
We provide sufficient conditions under which the difference of the resolvents of two higher-order operators acting in $R^N$ belongs to trace classes $cC^p$. We provide explicit estimates on the norm of the resolvent difference in terms of $L^p$ norms of the difference of the coefficients. Such inequalities are useful in estimating the effect of localized perturbations of the coefficients.
A conic bundle is a contraction $Xto Z$ between normal varieties of relative dimension $1$ such that $-K_X$ is relatively ample. We prove a conjecture of Shokurov which predicts that, if $Xto Z$ is a conic bundle such that $X$ has canonical singularities and $Z$ is $mathbb{Q}$-Gorenstein, then $Z$ is always $frac{1}{2}$-lc, and the multiplicities of the fibers over codimension $1$ points are bounded from above by $2$. Both values $frac{1}{2}$ and $2$ are sharp. This is achieved by solving a more general conjecture of Shokurov on singularities of bases of lc-trivial fibrations of relative dimension $1$ with canonical singularities.
We consider a family of spherically symmetric, asymptotically Euclidean manifolds with two trapped sets, one which is unstable and one which is semi-stable. The phase space structure is that of an inflection transmission set. We prove a sharp local smoothing estimate for the linear Schrodinger equation with a loss which depends on how flat the manifold is near each of the trapped sets. The result interpolates between the family of similar estimates in cite{ChWu-lsm}. As a consequence of the techniques of proof, we also show a sharp high energy resolvent estimate with a polynomial loss depending on how flat the manifold is near each of the trapped sets.
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