arXiv subjects

Richard Schoen

Publications and source records attributed to Richard Schoen.

At least 19 recordsLinked to original sources

arXiv2025

Free boundary minimal surfaces in products of balls

In this paper we develop an extremal eigenvalue approach to the problem of construction of free boundary minimal surfaces in the product of Euclidean balls of chosen radii. The extremal problem involves a linear combination of normalized mixed Steklov-Neumann eigenvalues. The problem is motivated by the Schwarz P-surface which is a free boundary minimal surface in a cube. We show that the problem often does not have an absolute maximum in the product case even though it is bounded from above. By imposing a finite group of symmetries on both the surface and on the eigenfunctions we construct at least one free boundary minimal surface in a rectangular prism with arbitrary side lengths. We further show that unless the rectangular prism is a cube there are at least two such surfaces. We also prove that any immersed free boundary minimal surface of genus 0 with one boundary component on each face of a rectangular prism, and that is invariant under the reflections interchanging opposite faces of the prism, is necessarily embedded. Finally we show that for a genus 0 surface with 6 boundary components and suitable reflection symmetries there is a maximizing metric which can be realized by a free boundary minimal immersion into a product of Euclidean balls.

math.DG↗

arXiv2023

On the first and second homotopy groups of manifolds with positive curvature

We show that an odd dimensional closed manifold with positive curvature cannot contain an incompressible real projective plane in the sense that there is no map of the projective plane into the manifold which is nontrivial on both first and second homotopy groups. Another way to say this is that no element of the fundamental group reverses the orientation of a class in the second homotopy group. Further we show that for a manifold of dimension divisible by four with positive curvature the fundamental group acts trivially on the second homotopy group. The methods involve a careful study of the stability of minimal two spheres in manifolds of positive curvature.

math.DG↗

arXiv2023

Stability and largeness properties of minimal surfaces in higher codimension

We consider stable minimal surfaces of genus 1 in Euclidean space and in Riemannian manifolds. Under the condition of covering stability (all finite covers are stable) we show that a genus 1 finite total curvature minimal surface in $\mathbb R^n$ lies in an even dimensional affine subspace and is holomorphic for some constant orthogonal complex structure. For stable minimal tori in Riemannian manifolds we give an explicit bound on the systole in terms of a positive lower bound on the isotropic curvature. As an application we estimate the systole of noncyclic abelian subgroups of the fundamental group of PIC manifolds. This gives a new proof of the result of [5] that the fundamental cannot contain a noncyclic free abelian subgroup. The proofs apply the structure theory of holomorphic vector bundles over genus 1 Riemann surfaces developed by M. Atiyah [2].

math.DG↗

arXiv2019

Some results on higher eigenvalue optimization

In this paper we obtain several results concerning the optimization of higher Steklov eigenvalues both in two and higher dimensional cases. We first show that the normalized (by boundary length) $k$-th Steklov eigenvalue on the disk is not maximized for a smooth metric on the disk for $k\geq 3$. For $k=1$ the classical result of [W] shows that $σ_1$ is maximized by the standard metric on the round disk. For $k=2$ it was shown [GP1] that $σ_2$ is not maximized for a smooth metric. We also prove a local rigidity result for the critical catenoid and the critical Möbius band as free boundary minimal surfaces in a ball under $C^2$ deformations. We next show that the first $k$ Steklov eigenvalues are continuous under certain degenerations of Riemannian manifolds in any dimension. Finally we show that for $k\geq 2$ the supremum of the $k$-th Steklov eigenvalue on the annulus over all metrics is strictly larger that that over $S^1$-invariant metrics. We prove this same result for metrics on the Möbius band.

math.DG↗

arXiv2018

Uniqueness theorems for free boundary minimal disks in space forms

We show that a minimal disk satisfying the free boundary condition in a constant curvature ball of any dimension is totally geodesic. We weaken the condition to parallel mean curvature vector in which case we show that the disk lies in a three dimensional constant curvature submanifold and is totally umbilic. These results extend to higher dimensions earlier three dimensional work of J. C. C. Nitsche and R. Souam.

math.DG↗

arXiv2017

Complete manifolds with bounded curvature and spectral gaps

We study the spectrum of complete noncompact manifolds with bounded curvature and positive injectivity radius. We give general conditions which imply that their essential spectrum has an arbitrarily large finite number of gaps. In particular, for any noncompact covering of a compact manifold, there is a metric on the base so that the lifted metric has an arbitrarily large finite number of gaps in its essential spectrum. Also, for any complete noncompact manifold with bounded curvature and positive injectivity radius we construct a metric uniformly equivalent to the given one (also of bounded curvature and positive injectivity radius) with an arbitrarily large finite number of gaps in its essential spectrum.

math.SP↗

arXiv2017

Shape optimization for the Steklov problem in higher dimensions

We show that the ball does not maximize the first nonzero Steklov eigenvalue among all contractible domains of fixed boundary volume in $\mathbb{R}^n$ when $n \geq 3$. This is in contrast to the situation when $n=2$, where a result of Weinstock from 1954 shows that the disk uniquely maximizes the first Steklov eigenvalue among all simply connected domains in the plane having the same boundary length. When $n \geq 3$, we show that increasing the number of boundary components does not increase the normalized (by boundary volume) first Steklov eigenvalue. This is in contrast to recent results which have been obtained for surfaces.

math.SP↗

arXiv2017

Sharp eigenvalue bounds and minimal surfaces in the ball

We prove existence and regularity of metrics on a surface with boundary which maximize sigma_1 L where sigma_1 is the first nonzero Steklov eigenvalue and L the boundary length. We show that such metrics arise as the induced metrics on free boundary minimal surfaces in the unit ball B^n for some n. In the case of the annulus we prove that the unique solution to this problem is the induced metric on the critical catenoid, the unique free boundary surface of revolution in B^3. We also show that the unique solution on the Mobius band is achieved by an explicit S^1 invariant embedding in B^4 as a free boundary surface, the critical Mobius band. For oriented surfaces of genus 0 with arbitrarily many boundary components we prove the existence of maximizers which are given by minimal embeddings in B^3. We characterize the limit as the number of boundary components tends to infinity to give the asymptotically sharp upper bound of 4pi. We also prove multiplicity bounds on sigma_1 in terms of the topology, and we give a lower bound on the Morse index for the area functional for free boundary surfaces in the ball.

math.DG↗

arXiv2017

Positive Scalar Curvature and Minimal Hypersurface Singularities

In this paper we develop methods to extend the minimal hypersurface approach to positive scalar curvature problems to all dimensions. This includes a proof of the positive mass theorem in all dimensions without a spin assumption. It also includes statements about the structure of compact manifolds of positive scalar curvature extending the work of \cite{sy1} to all dimensions. The technical work in this paper is to construct minimal slicings and associated weight functions in the presence of small singular sets and to show that the singular sets do not become too large in the lower dimensional slices. It is shown that the singular set in any slice is a closed set with Hausdorff codimension at least three. In particular for arguments which involve slicing down to dimension $1$ or $2$ the method is successful. The arguments can be viewed as an extension of the minimal hypersurface regularity theory to this setting of minimal slicings.

math.DG↗

arXiv2015

The spacetime positive mass theorem in dimensions less than eight

We prove the spacetime positive mass theorem in dimensions less than eight. This theorem states that for any asymptotically flat initial data set satisfying the dominant energy condition, the ADM energy-momentum vector $(E,P)$ of the initial data satisfies the inequality $E \ge |P|$. Previously, this theorem was proven only for spin manifolds by E. Witten. Our proof is a modification of the minimal hypersurface technique that was used by the last named author and S.-T. Yau to establish the time-symmetric case. Instead of minimal hypersurfaces, we use marginally outer trapped hypersurfaces (MOTS) whose existence is guaranteed by earlier work of the first named author. An important part of our proof is to introduce an appropriate substitute for the area functional used in the time-symmetric case. We also establish a density theorem of independent interest that allows us to reduce the general case of our theorem to the case of initial data that has harmonic asymptotics and satisfies the strict dominant energy condition.

math.DG↗

arXiv2015

Localizing solutions of the Einstein constraint equations

We perform an optimal localization of asymptotically flat initial data sets and construct data that have positive ADM mass but are exactly trivial outside a cone of arbitrarily small aperture. The gluing scheme that we develop allows to produce a new class of $N$-body solutions for the Einstein equation, which patently exhibit the phenomenon of gravitational shielding: for any large $T$ we can engineer solutions where any two massive bodies do not interact at all for any time $t\in(0,T)$, in striking contrast with the Newtonian gravity scenario.

math.DG↗

arXiv2015

On the Bartnik mass of apparent horizons

In this paper we characterize the intrinsic geometry of apparent horizons (outermost marginally outer trapped surfaces) in asymptotically flat spacetimes; that is, the Riemannian metrics on the two sphere which can arise. Furthermore we determine the minimal ADM mass of a spacetime containing such an apparent horizon. The results are conveniently formulated in terms of the quasi-local mass introduced by Bartnik in 1989. The Hawking mass provides a lower bound for Bartnik's quasilocal mass on apparent horizons by way of Penrose's conjecture on time symmetric slices, proven in 1997 by Huisken and Ilmanen and in full generality in 1999 by Bray. We compute Bartnik's mass for all non-degenerate apparent horizons and show that it coincides with the Hawking mass. As a corollary we disprove a conjecture due to Gibbons in the spirit of Thorne's hoop conjecture, and construct a new large class of examples of apparent horizons with the integral of the negative part of the Gauss curvature arbitrarily large.

math.DG↗

arXiv2013

Minimal surfaces and eigenvalue problems

We show that metrics that maximize the k-th Steklov eigenvalue on surfaces with boundary arise from free boundary minimal surfaces in the unit ball. We prove several properties of the volumes of these minimal submanifolds. For free boundary minimal submanifolds in the ball we show that the boundary volume is reduced up to second order under conformal transformations of the ball. For two-dimensional stationary integer multiplicity rectifiable varifolds that are stationary for deformations that preserve the ball, we prove that the boundary length is reduced under conformal transformations. We also give an overview of some of the known results on extremal metrics of the Laplacian on closed surfaces, and give a survey of our recent results from [FS2] on extremal metrics for Steklov eigenvalues on compact surfaces with boundary.

math.DG↗

arXiv2012

Convexity of reduced energy and mass angular momentum inequalities

In this paper, we extend the work in \cite{D}\cite{ChrusLiWe}\cite{ChrusCo}\cite{Co}. We weaken the asymptotic conditions on the second fundamental form, and we also give an $L^{6}-$norm bound for the difference between general data and Extreme Kerr data or Extreme Kerr-Newman data by proving convexity of the renormalized Dirichlet energy when the target has non-positive curvature. In particular, we give the first proof of the strict mass/angular momentum/charge inequality for axisymmetric Einstein/Maxwell data which is not identical with the extreme Kerr-Newman solution.

math.DG↗

arXiv2011

Specifying angular momentum and center of mass for vacuum initial data sets

We show that it is possible to perturb arbitrary vacuum asymptotically flat spacetimes to new ones having exactly the same energy and linear momentum, but with center of mass and angular momentum equal to any preassigned values measured with respect to a fixed affine frame at infinity. This is in contrast to the axisymmetric situation where a bound on the angular momentum by the mass has been shown to hold for black hole solutions. Our construction involves changing the solution at the linear level in a shell near infinity, and perturbing to impose the vacuum constraint equations. The procedure involves the perturbation correction of an approximate solution which is given explicitly.

math.DG↗

arXiv2010

The first Steklov eigenvalue, conformal geometry, and minimal surfaces

We consider the relationship of the geometry of compact Riemannian manifolds with boundary to the first nonzero eigenvalue sigma_1 of the Dirichlet-to-Neumann map (Steklov eigenvalue). For surfaces Sigma with genus gamma and k boundary components we obtain the upper bound sigma_1L(\partial Σ) \leq 2(2gamma+k)π. We attempt to find the best constant in this inequality for annular surfaces (gamma=0 and k=2). For rotationally symmetric metrics we show that the best constant is achieved by the induced metric on the portion of the catenoid centered at the origin which meets a sphere orthogonally and hence is a solution of the free boundary problem for the area functional in the ball. For a general class of (not necessarily rotationally symmetric) metrics on the annulus, which we call supercritical, we prove that $σ_1(\sig)L(\pΣ)$ is dominated by that of the critical catenoid with equality if and only if the annulus is conformally equivalent to the critical catenoid by a conformal transformation which is an isometry on the boundary. We prove general upper bounds for conformal metrics on manifolds of any dimension which can be properly conformally immersed into the unit ball in terms of certain conformal volume quantities. We show that these bounds are only achieved when the manifold is minimally immersed by first Steklov eigenfunctions. We also use these ideas to show that any free boundary solution in two dimensions has area at least π.

math.DG↗

arXiv2010

On the existence of Hamiltonian stationary Lagrangian submanifolds in symplectic manifolds

Let (M,w) be a compact symplectic 2n-manifold, and g a Riemannian metric on M compatible with w. For instance, g could be Kahler, with Kahler form w. Consider compact Lagrangian submanifolds L of M. We call L Hamiltonian stationary, or H-minimal, if it is a critical point of the volume functional under Hamiltonian deformations. It is called Hamiltonian stable if in addition the second variation of volume under Hamiltonian deformations is nonnegative. Our main result is that if L is a compact, Hamiltonian stationary Lagrangian in C^n satisfying the extra condition of being Hamiltonian rigid, then for any M,w,g as above there exist compact Hamiltonian stationary Lagrangians L' in M contained in a small ball about some p in M and locally modelled on tL for small t>0, identifying M near p with C^n near 0. If L is Hamiltonian stable, we can take L' to be Hamiltonian stable. Applying this to known examples L in C^n shows that there exist families of Hamiltonian stable, Hamiltonian stationary Lagrangians diffeomorphic to T^n, and to (S^1 x S^{n-1})/{1,-1}, and with other topologies, in every compact symplectic 2n-manifold (M,w) with compatible metric g.

math.DG↗

arXiv2005

A generalization of Hawking's black hole topology theorem to higher dimensions

Hawking's theorem on the topology of black holes asserts that cross sections of the event horizon in 4-dimensional asymptotically flat stationary black hole spacetimes obeying the dominant energy condition are topologically 2-spheres. This conclusion extends to outer apparent horizons in spacetimes that are not necessarily stationary. In this paper we obtain a natural generalization of Hawking's results to higher dimensions by showing that cross sections of the event horizon (in the stationary case) and outer apparent horizons (in the general case) are of positive Yamabe type, i.e., admit metrics of positive scalar curvature. This implies many well-known restrictions on the topology, and is consistent with recent examples of five dimensional stationary black hole spacetimes with horizon topology $S^2 \times S^1$. The proof is inspired by previous work of Schoen and Yau on the existence of solutions to the Jang equation (but does not make direct use of that equation).

gr-qc↗