An official website of the United States government
Abstract We give a necessary condition on a geodesic in a Riemannian manifold that can run in some convex hypersurface.As a corollary, we obtain peculiar properties that hold true for every convex set in any generic Riemannian manifold ( M , g ) {(M,g)} .For example, if a convex set in ( M , g ) {(M,g)} is bounded by a smooth hypersurface, then it is strictly convex.
more »
« less
The p-widths of a surface
Abstract The $$p$$ p -widths of a closed Riemannian manifold are a nonlinear analogue of the spectrum of its Laplace–Beltrami operator, which corresponds to areas of a certain min-max sequence of possibly singular minimal submanifolds. We show that the $$p$$ p -widths of any closed Riemannian two-manifold correspond to a union of closed immersed geodesics, rather than simply geodesic nets. We then prove optimality of the sweepouts of the round two-sphere constructed from the zero set of homogeneous polynomials, showing that the $$p$$ p -widths of the round sphere are attained by $$\lfloor \sqrt{p}\rfloor $$ ⌊ p ⌋ great circles. As a result, we find the universal constant in the Liokumovich–Marques–Neves–Weyl law for surfaces to be $$\sqrt{\pi }$$ π . En route to calculating the $$p$$ p -widths of the round two-sphere, we prove two additional new results: a bumpy metrics theorem for stationary geodesic nets with fixed edge lengths, and that, generically, stationary geodesic nets with bounded mass and bounded singular set have Lusternik–Schnirelmann category zero.
more »
« less
- PAR ID:
- 10437025
- Date Published:
- Journal Name:
- Publications mathématiques de l'IHÉS
- Volume:
- 137
- Issue:
- 1
- ISSN:
- 0073-8301
- Page Range / eLocation ID:
- 245 to 342
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
We study an inverse problem of determining a time-dependent damping coefficient and potential appearing in the wave equation in a compact Riemannian manifold of dimension three or higher. More specifically, we are concerned with the case of conformally transversally anisotropic manifolds, or in other words, compact Riemannian manifolds with boundary conformally embedded in a product of the Euclidean line and a transversal manifold. With an additional assumption of the attenuated geodesic ray transform being injective on the transversal manifold, we prove that the knowledge of a certain partial Cauchy data set determines the time-dependent damping coefficient and potential uniquely.more » « less
-
It is well-known that under suitable hypotheses, for a sequence of solutions of the (simplified) Ginzburg–Landau equations-\Delta u_{\varepsilon} +\varepsilon^{-2}(|u_{\varepsilon}|^{2}-1)u_{\varepsilon} = 0, the energy and vorticity concentrate as\varepsilon\to 0around a codimension2stationary varifold – a (measure-theoretic) minimal surface. Much less is known about the question of whether, given a codimension2minimal surface, there exists a sequence of solutions for which the given minimal surface is the limiting concentration set. The corresponding question is very well-understood for minimal hypersurfaces and the scalar Allen–Cahn equation, and for the Ginzburg–Landau equations when the minimal surface is locally area-minimizing, but otherwise quite open. We consider this question on a3-dimensional closed Riemannian manifold(M,g), and we prove that any embedded nondegenerate closed geodesic can be realized as the asymptotic energy/ vorticity concentration set of a sequence of solutions of the Ginzburg–Landau equations.more » « less
-
We study an inverse problem of determining a time-dependent potential appearing in the wave equation on conformally transversally anisotropic manifolds of dimension three or higher. These are compact Riemannian manifolds with boundary that are conformally embedded in a product of the real line and a transversal manifold. Under the assumption of the attenuated geodesic ray transform being injective on the transversal manifold, we prove the unique determination of time-dependent potentials from the knowledge of a certain partial Cauchy data set.more » « less
-
Abstract We note that the recent polynomial proofs of the spherical and complex plank covering problems by Zhao and Ortega-Moreno give some general information on zeros of real and complex polynomials restricted to the unit sphere. As a corollary of these results, we establish several generalizations of the celebrated Bang plank covering theorem. We prove a tight polynomial analog of the Bang theorem for the Euclidean ball and an even stronger polynomial version for the complex projective space. Specifically, for the ball, we show that for every real nonzero $$d$$-variate polynomial $$P$$ of degree $$n$$, there exists a point in the unit $$d$$-dimensional ball at distance at least $1/n$ from the zero set of the polynomial $$P$$. Using the polynomial approach, we also prove the strengthening of the Fejes Tóth zone conjecture on covering a sphere by spherical segments, closed parts of the sphere between two parallel hyperplanes. In particular, we show that the sum of angular widths of spherical segments covering the whole sphere is at least $$\pi $$.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1007/s10240-023-00141-7
