Eigenvalue problems and free boundary minimal surfaces in spherical caps

We study spaces and moduli spaces of Riemannian metrics with non-negative Ricci or non-negative sectional curvature on closed and open manifolds. We construct, in particular, the first classes of manifolds for which these moduli spaces have non-trivial rational homotopy, homology and cohomology groups. We also show that in every dimension at least seven (respectively, at least eight) there exist infinite sequences of closed (respectively, open) manifolds of pairwise distinct homotopy type for which the space and moduli space of Riemannian metrics with non-negative sectional curvature has infinitely many path components. A completely analogous statement holds for spaces and moduli spaces of non-negative Ricci curvature metrics.

We classify the affine connections on compact orientable surfaces for which the pseudogroup of local isometries acts transitively. We prove that such a connection is either torsion-free and flat, the Levi–Civita connection of a Riemannian metric of constant curvature or the quotient of a translation-invariant connection in the plane. This refines previous results by Opozda.

We show that, on an oriented compact surface, two sufficiently $C^{2}$-close Riemannian metrics with strictly convex boundary, no conjugate points, hyperbolic trapped set for their geodesic flows and the same marked boundary distance are isometric via a diffeomorphism that fixes the boundary. We also prove that the same conclusion holds on a compact surface for any two negatively curved Riemannian metrics with strictly convex boundary and the same marked boundary distance, extending a result of Croke and Otal.

We study Gauss curvature for random Riemannian metrics on a compact surface,lying in a fixed conformal class; our questions are motivated by comparisongeometry. We next consider analogous questions for the scalar curvature indimension $n>2$, and for the $Q$-curvature of random Riemannian metrics.

Surfaces with the same marked length spectrum

Conformal equivalence theorem from complex analysis says that every Riemannian metric on a compact surface with negative Euler characteristics can be obtained by multiplying a metric of constant negative curvature by a scalar function. This fact is used to produce information about the topological and metric entropies of the geodesic flow associated with a Riemannian metric, geodesic length spectrum, geodesic and harmonic measures of infinity and Cheeger asymptotic isoperimetric constant. The method is rather uniform and is based on a comparison of extremals for variational problems for conformally equivalent metrics.

We establish in a canonical manner a manifold structure for the completed space of bounded maps between open manifoldsM andN, assuming thatM andN are endowed with Riemannian metrics of bounded geometry up to a certain order. The identity component of the corresponding diffeomorphisms is a Banach manifold and metrizable topological group.

Conformal deformation of warped products and scalar curvature functions on open manifolds

In this chapter we will describe results about spaces and moduli spaces of complete Riemannian metrics with non-negative sectional curvature on open manifolds. A new and important tool for understanding these spaces involves employing properties of the so-called ‘souls’ of the metrics, and we start with a discussion of these.KeywordsModulus SpaceVector BundleLens SpaceEuler ClassRiemannian SubmersionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

A new flow on an open manifold with bounded geometry has been considered. We prove short time existence and uniqueness for the flow. The approach is based on the study of the geometry of the manifold of Riemannian metrics on open manifolds. Moreover, some properties of this flow are investigated.

This paper is concerned with the study of the wave equation on compact surfaces and locally distributed damping, described byutt−ΔMu+a(x)g(ut)=0on \thinspace M×]0,∞[,\begin{equation} \left . \begin {array}{l} u_{tt} - \Delta _{\mathcal {M}}u+ a(x) g(u_{t})=0 \; \text {on \thinspace }\mathcal {M}\times \left ] 0,\infty \right [ , \end{array} \right . \nonumber \end{equation}whereM⊂R3\mathcal {M}\subset \mathbb {R}^3is a smooth oriented embedded compact surface without boundary. Denoting byg\mathbf {g}the Riemannian metric induced onM\mathcal {M}byR3\mathbb {R}^3, we prove that for eachϵ>0\epsilon > 0, there exist an open subsetV⊂MV \subset \mathcal Mand a smooth functionf:M→Rf:\mathcal M \rightarrow \mathbb Rsuch thatmeas(V)≥meas(M)−ϵmeas(V)\geq meas(\mathcal M)-\epsilon,Hessf≈gHess f \approx \mathbf {g}onVVandinfx∈V|∇f(x)|>0\underset {x\in V}\inf |\nabla f(x)|>0.In addition, we prove that ifa(x)≥a0>0a(x) \geq a_0> 0on an open subsetM∗⊂M\mathcal {M}{\ast } \subset \mathcal Mwhich containsM∖V\mathcal {M}\backslash Vand ifggis a monotonic increasing function such thatk|s|≤|g(s)|≤K|s|k |s| \leq |g(s)| \leq K |s|for all|s|≥1|s| \geq 1, then uniform and optimal decay rates of the energy hold.

In this article, we prove that on any compact Riemann surface (M,∂M,g) with non-empty smooth boundary ∂M and a Riemannian metric g, (i) any K∈C∞(M) is the Gaussian curvature function of some Riemannian metric on M; (ii) any σ∈C∞(∂M) is the geodesic curvature of some Riemannian metric on M. These geometric results are obtained analytically by solving a semi-linear elliptic equation −Δgu=Ke2u on M with oblique boundary condition ∂u∂ν=σeu . One essential tool is the existence results of Brezis–Merle type equations −Δgu+Au=Ke2uinM and ∂u∂ν+κu=σeuon∂M with given functions K,σ and some constants A,κ . In addition, we rely on the extension of the uniformization theorem given by Osgood, Phillips and Sarnak.

Let V ⊂ R N be a compact real analytic surface with isolated singularities, and assume its smooth part V 0 is equipped with a Riemannian metric that is induced from some analytic Riemannian metric on R N . We prove: (1) Each point of V has a neighborhood which is quasi-isometric (naturally and almost isometrically ) to a union of metric cones and horns, glued at their tips. (2) A full asymptotic expansion, for any p ∈ V, of the length of V n {q: dist (q, p) = r} as r → 0. 0. (3) A Gauss-Bonnet Theorem, saying that each singular point contributes 1 - l/(2π), where l is the coefficient of the linear term in the expansion of (2). (4) The L 2 Stokes Theorem, selfadjointness and discreteness of the Laplace-Beltrami operator on Vo, an estimate on the heat kernel, and a Gauss-Bonnet Theorem for the L 2 Euler characteristic. As a central tool we use resolution of singularities.

We consider geodesic flows of Riemannian metrics on T 2 and S 2 integrable by an integral depending on the momenta linearly or quadratically (in what follows such integrals are referred to as linear and quadratic integrals) and show that there exists an effective criterion for topological equivalence of two arbitrary flows. The criterion is based on comparing the "codes" of the corresponding Riemannian metrics (see below). A. T. Fomenko [1, 2] found a new topological invariant I(H, Q) of integrable Hamiltonian systems, making it possible to classify integrable Hamiltonians up to the coarse topological equivalence. In this case the labeled topological invariant I*(H, Q) discovered by A. T. Fomenko and H. Wsishang [3] (also called the labeled molecule [4]) classifies integrable Hamiltonians up to a finer topological equivalence. Let M 4 be a smooth symplectic manifold, and let v = sgrad H be an integrable Hamiltonian system with smooth Hamiltonian H . Following [1] we say that an integral F on an isoenergetic surface Q3 = {H = const} is a Bott integral if all its critical points form nondegenerate critical submanifolds. Definition (see [4]). A pair (P1, K1), where P1 is a compact orientable surface and K1 a nonempty finite connected graph on P1, is called a letter-atom if the following conditions hold: 1) the degree of each vertex of 1(1 is equal to 0, 2, or 4; 2) each connected component of the difference P1 \ K1 is homeomorphic to the annulus S 1 x (0, 1]; 3) the set of the annuli forming P1 \ :K~ can be divided into two parts (positive and negative annuli) so that exactly one positive and one negative annulus adjoin each edge of K1. A molecule W, which is a union of atoms, is a pair ( P , K ) , where K is a graph embedded in a two-dimensional surface p2. A labeled molecule W* is obtained by supplementing W with numerical (rational or integer) labels. To each unlabeled molecule W we assign its complexity (m, n) , where m is the number of vertices in K (coinciding with the total number of critical circles of the Bott integral) and n is the number of connected components of P \ K (or the number of cylinders S 1 x S 1 x (0, 1) into which the isoenergetic manifold Q3 splits On deleting all singular leaves of the corresponding Liouville foliation). Fomenko posed the problem of describing all labeled molecules W* for the above type of integrable geodesic flows on T 2 and S 2 and of finding the corresponding (m, n)-domain in the molecular complexity table. For the torus T 2 this problem has been completely investigated by Selivanova whose results, as it turned out later, can be applied effectively to the topological classification of geodesic flows on the sphere S 2 with nontrivial quadratic integral. Here we also consider geodesic flows on S 2 with linear integral and thus complete the investigation of integrable geodesic flows on two-dimensional Riemannian manifolds (with additional integral of degree no higher than 2). As is known [6, 9], the geodesic flows on two-dimensional surfaces of genus > 1 have no additional integrals analytic with respect to the momenta. In §2 we give the basic facts concerning the above type of geodesic flows that are used in the subsequent presentation and are based on the results of Darboux [10], Levi-Civita [11], Birkhoff [5], and Kolokol'tsov

A complete noncompact manifold M with nonnegative sectional curvature is diffeomorphic to the normal bundle of a compact submanifold S called the soul of M . When S is a round sphere we show that the clutching map of this bundle is restricted; this is used to deduce that there are at most finitely many isomorphism types of such bundles with sectional curvature lying in a fixed interval [0, κ]. We also examine the opposite question of how the twisting of the bundle limits the type of possible nonnegative curvature metrics on the bundle: It turns out that if the bundle does not admit a nowhere-zero section, then the normal exponential map is necessarily a diffeomorphism onto M , and the ideal boundary of M consists of a single point. In their paper [CG], Cheeger and Gromoll raised the question of which vector bundles over the round sphere admit complete metrics with nonnegative sectional curvature. The significance of this problem is that it attempts to determine to what extent a converse to the Soul theorem holds. Recall that this theorem states that every open (i.e., complete noncompact) manifold M with nonnegative curvature KM is diffeomorphic to a vector bundle over a compact totally geodesic submanifold S called a soul. A natural question then is whether all such vector bundles admit complete metrics with KM ≥ 0. In [OW], it was shown that when the soul is a Bieberbach manifold, nonnegative curvature metrics force the vanishing of the Euler class of the vector bundle. It follows that among oriented plane bundles over the torus, only the trivial one admits such a metric. The above case is fairly rigid a priori, however (since for example any bundle with curvature ≥ 0 over such a manifold must also admit a flat metric), and the corresponding question for a simply connected base remains open. An answer could provide insights on the topological structure of compact manifolds with nonnegative curvature, since any open manifold with nonnegative curvature can (after modifying its metric) be isometrically 1991 Mathematics Subject Classification. Primary 53C20.