Related papers: Lower bounds for eigenfunction restrictions in lac…

Let $(M,g)$ be a compact Riemannian surface with nonpositive sectional curvature and let $\gamma$ be a closed geodesic in $M$. And let $e_\lambda$ be an $L^2$-normalized eigenfunction of the Laplace-Beltrami operator $\Delta_g$ with…

In quantum theory on curved backgrounds, Heisenberg's uncertainty principle is usually discussed in terms of ensemble variances and flat-space commutators. Here we take a different, preparation-based viewpoint tailored to sharp position…

Let $(M,g)$ be a smooth, compact, Riemannian manifold and $\{\phi_h\}$ a sequence of $L^2$-normalized Laplace eigenfunctions on $M$. For a smooth submanifold $H\subset M$, we consider the growth of the restricted eigenfunctions $\phi_h|_H$…

A 7-dimensional area-minimizing embedded hypersurface $M$ will in general have a discrete singular set. The same is true if $M$ is stable, or has bounded index, provided $H^6(sing M) = 0$. We show that if $M_i$ are a sequence of such…

We show how Lasry-Lions's result on regularization of functions defined on $\mathbb{R}^n$ or on Hilbert spaces by sup-inf convolutions with squares of distances can be extended to (finite or infinite dimensional) Riemannian manifolds $M$ of…

We derive upper bounds on the difference between the orthogonal projections of a smooth function $u$ onto two finite element spaces that are nearby, in the sense that the support of every shape function belonging to one but not both of the…

We study curvature functionals for immersed 2-spheres in non-compact, three-dimensional Riemannian manifold $(M,h)$ without boundary. First, under the assumption that $(M,h)$ is the euclidean 3-space endowed with a semi-perturbed metric…

Let $(M, g)$ be a compact real analytic Riemannian manifold and $\pi \colon \widetilde{M} \to M$ its universal cover. Assume that $\widetilde{M}$ can be realised as a manifold definable in an o-minimal structure $\Sigma$ expanding…

Let $M$ be a compact smooth manifold equipped with a positive smooth density $\mu$ and $H$ be a smooth distribution endowed with a fiberwise inner product $g$. We define the Laplacian $\Delta_H$ associated with $(H,\mu,g)$ and prove that it…

We prove that a deformation of a hypersurface in a $(n+1)$-dimensional real space form ${\mathbb S}^{n+1}_{p,1}$ induce a Hamiltonian variation of the normal congruence in the space ${\mathbb L}({\mathbb S}^{n+1}_{p,1})$ of oriented…

Let $(M^{n+1},g,e^{-f}d\mu)$ be a complete smooth metric measure space with $2\leq n\leq 6$ and Bakry-\'{E}mery Ricci curvature bounded below by a positive constant. We prove a smooth compactness theorem for the space of complete embedded…

We prove that if a family of metrics, $g_i$, on a compact Riemannian manifold, $M^n$, have a uniform lower Ricci curvature bound and converge to $g_\infty$ smoothly away from a singular set, $S$, with Hausdorff measure, $H^{n-1}(S) = 0$,…

Given a compact Riemannian manifold $(M, g)$ without boundary, we estimate the Lebesgue norm of Laplace-Beltrami eigenfunctions when restricted to a wide variety of subsets $\Gamma$ of $M$. The sets $\Gamma$ that we consider are Borel…

In compact Riemannian manifolds of dimension 3 or higher with positive Ricci curvature, we prove that every constant mean curvature hypersurface produced by the Allen-Cahn min-max procedure of Bellettini-Wickramasekera (with constant…

We prove quantitative unique continuation results for the semiclassical Schrodinger operator on smooth, compact domains. These take the form of exponentially decreasing (in h) local L^{2} lower bounds for exponentially precise quasimodes.…

Let $(X, \omega)$ be a compact symplectic manifold and $L$ be a Lagrangian submanifold. Suppose $(X, L)$ has a Hamiltonian $S^1$ action with moment map $\mu$. Take an invariant $\omega$-compatible almost complex structure, we consider…

We address the problem of integrability of the sub-Riemannian mean curvature of an embedded hypersurface around isolated characteristic points. The main contribution of this note is the introduction of a concept of mildly degenerate…

If $(M,g)$ is a compact Riemannian manifold of dimension $n\ge 2$ we give necessary and sufficient conditions for improved $L^p(M)$-norms of eigenfunctions for all $2<p\ne p_c=\tfrac{2(n+1)}{n-1}$, the critical exponent. Since improved…

The problem of obtaining the lower bounds on the restriction of Laplacian eigenfunctions to hypersurfaces inside a compact Riemannian manifold $(M,g)$ is challenging and has been attempted by many authors \cite{BR, GRS, Jun, ET}. This paper…

We obtain quasimode, eigenfunction and spectral projection bounds for Schr\"odinger operators, $H_V=-\Delta_g+V(x)$, on compact Riemannian manifolds $(M,g)$ of dimension $n\ge2$, which extend the results of the third author~\cite{sogge88}…