Blow up solutions of an asymptotically critical problem involving fractional Laplacian in compact Riemannian manifold

We study compact Riemannian manifolds for which the light between any pair of points is blocked by finitely many point shades. Compact flat Riemannian manifolds are known to have this finite blocking property. We conjecture that amongst compact Riemannian manifolds this finite blocking property characterizes the flat metrics. Using entropy considerations, we verify this conjecture amongst metrics with nonpositive sectional curvatures. Using the same approach, K. Burns and E. Gutkin have independently obtained this result. Additionally, we show that compact quotients of Euclidean buildings have the finite blocking property. On the positive curvature side, we conjecture that compact Riemannian manifolds with the same blocking properties as compact rank one symmetric spaces are necessarily isometric to a compact rank one symmetric space. We include some results providing evidence for this conjecture.

In this paper we explain how the so-called adapted complex structures can be used to associate to each compact real-analytic Riemannian manifold a family of complete Kahler-Einstein metrics and show that already one element of this family uniquely determines the original manifold. The underlying manifolds of these metrics are open disc bundles in the tangent bundle of the original Riemannian manifold. Recall first the notion of adapted complex structures (cf. [7], [12], [16]). Let (M, g) be a complete real-analytic Riemannian manifold. If γ is a geodesic in M , we can define a map ψγ : C→ TM by ψγ : σ + iτ 7→ τ γ(σ). If r ∈ (0,∞], we let T M denote the open disc bundle in TM consisting of tangent vectors of norm less than r (note that we allow r to be infinite). A complex structure on T M is said to be adapted with respect to g if ψγ is holomorphic on ψ−1 γ (T M) for each geodesic γ. We shall usually omit the phrase “with respect to” if it is obvious which metric is being discussed. The manifolds T M are also called Grauert tubes since Grauert used such manifolds in his famous result to show that each real-analytic manifold admits a real-analytic embedding to a euclidean space. Adapted complex structures were discovered by studying certain global solutions of the complex homogeneous Monge-Ampere equation on Stein manifolds (cf. [2], [7], [12], [14]). Their basic properties were treated in [7], [12], [16]. Among others, one has an existence result: if (M, g) is a compact real-analytic Riemannian manifold, then there exists an r ∈ (0,∞] such that T M carries an adapted complex structure [7], [16]. Also the adapted complex structure is uniquely determined by (M, g) ([7], [12]). From now on we shall take (M, g) to be a compact Riemannian manifold and use R to denote the largest element of (0,∞] such that TM supports an adapted complex structure. It was shown in [12] that the energy function E on TM which assigns to each tangent vector the half of its norm-square with respect to the metric g is strictly plurisubharmonic on TM . Thus another theorem of Grauert implies that TM is a Stein manifold. Hence T M is relatively compact and strictly pseudoconvex whenever 0 < r < R. Received by the editors February 2, 2000. 2000 Mathematics Subject Classification. Primary 32Q15, 53C35.

The purpose of this paper is to show well-posedness results for Dirichlet problems for the Stokes and Navier–Stokes systems with $$L^{\infty }$$ -variable coefficients in $$L^2$$ -based Sobolev spaces in Lipschitz domains on compact Riemannian manifolds. First, we refer to the Dirichlet problem for the nonsmooth coefficient Stokes system on Lipschitz domains in compact Riemannian manifolds and show its well-posedness by employing a variational approach that reduces the boundary value problem of Dirichlet type to a variational problem defined in terms of two bilinear continuous forms, one of them satisfying a coercivity condition and another one the inf-sup condition. We show also the equivalence between some transmission problems for the nonsmooth coefficient Stokes system in complementary Lipschitz domains on compact Riemannian manifolds and their mixed variational counterparts, and then their well-posedness in $$L^2$$ -based Sobolev spaces by using the remarkable Necas–Babuska–Brezzi technique (see Babuska in Numer Math 20:179–192, 1973; Brezzi in RAIRO Anal Numer R2:129–151, 1974; Necas in Rev Roum Math Pures Appl 9:47–69, 1964). As a consequence of these well-posedness results we define the layer potential operators for the nonsmooth coefficient Stokes system on Lipschitz surfaces in compact Riemannian manifolds, and provide their main mapping properties. These properties are used to construct explicitly the solution of the Dirichlet problem for the Stokes system. Further, we combine the well-posedness of the Dirichlet problem for the nonsmooth coefficient Stokes system with a fixed point theorem to show the existence of a weak solution to the Dirichlet problem for the nonsmooth variable coefficient Navier–Stokes system in $$L^2$$ -based Sobolev spaces in Lipschitz domains on compact Riemannian manifolds. The well developed potential theory for the smooth coefficient Stokes system on compact Riemannian manifolds (cf. Dindos and Mitrea in Arch Ration Mech Anal 174:1–47, 2004; Mitrea and Taylor in Math Ann 321:955–987, 2001) is also discussed in the context of the potential theory developed in this paper.

It is well known that the spectrum of Laplacian on a compact Riemannian manifold M is an important analytic invariant and has important geometric meanings. There are many mathematicians to investigate properties of the spectrum of Laplacian and to estimate the spectrum in term of the other geometric quantities of M . When M is a bounded domain in Euclidean spaces, a compact homogeneous Riemannian manifold, a bounded domain in the standard unit sphere or a compact minimal submanifold in the standard unit sphere, the estimates of the k + 1 -th eigenvalue were given by the first k eigenvalues (see [9], [12], [19], [20], [22], [23], [24] and [25]). In this paper, we shall consider the eigenvalue problem of the Laplacian on compact Riemannian manifolds. First of all, we shall give a general inequality of eigenvalues. As its applications, we study the eigenvalue problem of the Laplacian on a bounded domain in the standard complex projective space C P n ( 4 ) and on a compact complex hypersurface without boundary in C P n ( 4 ) . We shall give an explicit estimate of the k + 1 -th eigenvalue of Laplacian on such objects by its first k eigenvalues.

The geometry of [formula omitted]-harmonic maps

1. The main result and some consequences. In 1956 E. Calabi [6] attacked the classification problem of compact euclidean space forms by means of a special construction, called the Calabi construction (see Wolf [14, p. 124]). Here we announce that the construction can be extended to compact riemannian manifolds whose Ricci curvature tensor is zero (Ricci flat). Of course, it is not known if there exist any Ricci flat nonflat compact riemannian manifolds, and in fact a search for such manifolds was the original motivation for our study. However, as a consequence of our extension of Calabi's result we reduce the question of existence of a compact nonflat Ricci flat manifold to the simply connected, connected case. In any case, we essentially reduce the construction of compact Ricci flat manifolds to the lower-dimensional case together with the case of first Betti number zero. As a further consequence of our construction we extend one of the Bieberbach theorems [4], [14, Theorem 3.3.1] from the flat to the Ricci flat case (Theorem 1.4) and give various sufficient topological conditions for a Ricci flat manifold to be flat. Our main result is the following:

LetM be a compact Riemannian manifold with smooth boundary ∂M. We get bounds for the first eigenvalue of the Dirichlet eigenvalue problem onM in terms of bounds of the sectional curvature ofM and the normal curvatures of ∂M. We discuss the equality, which is attained precisely on certain model spaces defined by J. H. Eschenburg. We also get analog results for Kahler manifolds. We show how the same technique gives comparison theorems for the quotient volume(P)/volume(M),M being a compact Riemannian or Kahler manifold andP being a compact real hypersurface ofM.

We study inverse boundary problems for magnetic Schr\"odinger operators on a compact Riemannian manifold with boundary of dimension $\ge 3$. In the first part of the paper we are concerned with the case of admissible geometries, i.e. compact Riemannian manifolds with boundary which are conformally embedded in a product of the Euclidean line and a simple manifold. We show that the knowledge of the Cauchy data on the boundary of the manifold for the magnetic Schr\"odinger operator with $L^\infty$ magnetic and electric potentials, determines the magnetic field and electric potential uniquely. In the second part of the paper we address the case of more general conformally transversally anisotropic geometries, i.e. compact Riemannian manifolds with boundary which are conformally embedded in a product of the Euclidean line and a compact manifold, which need not be simple. Here, under the assumption that the geodesic ray transform on the transversal manifold is injective, we prove that the knowledge of the Cauchy data on the boundary of the manifold for a magnetic Schr\"odinger operator with continuous potentials, determines the magnetic field uniquely. Assuming that the electric potential is known, we show that the Cauchy data determines the magnetic potential up to a gauge equivalence.

S being the Ricci tensor. While every manifold with parallel Ricci tensor has harmonic curvature, i.e., satisfies fiR=O, there are examples ([3], Theorem 5.2) of open Riemannian manifolds with fiR=O and VS+O. In [1] Bourguignon has asked the question whether the Ricci tensor of a compact Riemannian manifold with harmonic curvature must be parallel. The aim of this paper is to give examples (see Remark 2) answering this question in the negative. All our examples are conformally flat (Corollary 1). Moreover, we obtain some classification results, restricting our consideration to Riemannian manifolds with fiR = O, VS + 0 and such that the Ricci tensor S has at any point less than three distinct eigenvalues. Starting from a description of their local structure at generic points (Theorem 1), we find all four-dimensional, analytic, complete and simply connected manifolds of this type (Theorem 2). They are all non-compact, but some of them do possess compact quotients. Next we prove (Theorem 3) that all compact four-dimensional analytic Riemannian manifolds with the above properties are covered by S 1 x S 3 with a metric of an explicitly described form. Throughout this paper, by a manifold we mean a connected paracompact manifold of class C ~ or analytic. By abuse of notation, concerning Riemannian manifolds we often write M instead of (M,g) and @ , v ) instead of g(u,v) for tangent vectors u, v.

We prove the existence and the uniqueness of a solution to the stochastic NSLE on a two-dimensional compact riemannian manifold. Thus we generalize a recent work by Burq, G\'erard and Tzvetkov in the deterministic setting, and a series of papers by de Bouard and Debussche, who have examined similar questions in the case of the flat euclidean space with random perturbation. We prove the existence and the uniqueness of a local maximal solution to stochastic nonlinear Schr\"odinger equations with multiplicative noise on a compact d-dimensional riemannian manifold. Under more regularity on the noise, we prove that the solution is global when the nonlinearity is of defocusing or of focusing type, d=2 and the initial data belongs to the finite energy space. Our proof is based on improved stochastic Strichartz inequalities.

We study harmonic morphisms by placing them into the context of conformal foliations. Most of the results we obtain hold for fibres of dimension one and codomains of dimension not equal to two. We consider foliations which produce harmonic morphisms on both compact and noncompact Riemannian manifolds. By using integral formulae, we prove an extension to one-dimensional foliations which produce harmonic morphisms of the well-known result of S. Bochner concerning Killing fields on compact Riemannian manifolds with nonpositive Ricci curvature. From the noncompact case, we improve a result of R. L. Bryant[9] regarding harmonic morphisms with one-dimensional fibres defined on Riemannian manifolds of dimension at least four with constant sectional curvature. Our method gives an entirely new and geometrical proof of Bryant's result. The concept of homothetic foliation (or, more generally, homothetic distribution) which we introduce, appears as a useful tool both in proofs and in providing new examples of harmonic morphisms, with fibres of any dimension.

For any n-dimensional compact spin Riemannian manifold M with a given spin structure and a spinor bundle Σ M, and any compact Riemannian manifold N, we show an ϵ-regularity theorem for weakly Dirac-harmonic maps (ϕ, ψ):M ⊗ Σ M → N ⊗ ϕ * TN. As a consequence, any weakly Dirac-harmonic map is proven to be smooth when n = 2. A weak convergence theorem for approximate Dirac-harmonic maps is established when n = 2. For n ≥ 3, we introduce the notation of stationary Dirac-harmonic maps and obtain a Liouville theorem for stationary Dirac-harmonic maps in . If, in addition, ψ ∈ W 1,p for some , then we obtain an energy monotonicity formula and prove a partial regularity theorem for any such a stationary Dirac-harmonic map.

Several invariant tests for uniformity of a distribution on the circle, the sphere and the hemisphere have been proposed by Rayleigh, Watson, Bingham, Ajne, Beran and others. In this paper a class of invariant tests for uniformity on compact Riemannian manifolds containing many of the known ones is presented and studied (the asymptotic theory as well as some local optimality properties for this class of tests are given). The examples include two new tests, one for the sphere and the other for the hemisphere. Let $X$ be a compact Riemannian manifold, $\mu$ the normalized volume element (the uniform distribution of $X$) and $\nu_n$ the empirical distribution corresponding to a sequence of i.i.d. $X$-valued random variables. The statistics in which these tests are based are just convergent weighted sums of the squares of the Fourier coefficients of $\nu_n(\omega) - \mu$ with respect to any orthonormal basis of $L_2(X, \mu)$ consisting of eigenfunctions of the Laplacian. An additional condition is imposed on the weights, namely that weights corresponding to coefficients of eigenfunctions in the same eigenspace of the Laplacian be equal (this condition is essential for the invariance of the tests). These statistics are related to Sobolev norms and so, the tests are called Sobolev tests. In connection with Sobolev statistics, it is interesting to note that the Sobolev norms of index $-s, s > (\dim X)/2$, metrize the weak-star topology of $\mathscr{P}(X)$, the space of Borel probability measures on $X$. A theorem about weak convergence of empirical distributions on compact manifolds, useful in proving some of the asymptotic results for Sobolev statistics, is also included. One of the sections (Section 2) is almost entirely devoted to give a short review of the facts needed in the paper about Riemannian manifolds, the Laplacian and Sobolev spaces.

We introduce the Local Increasing Regularity Method (LIRM) which allows us to get from \emph{local} a priori estimates, on solutions $u$ of a linear equation $\displaystyle Du=\omega ,$ \emph{global} ones. As an application we shall prove that if $D$ is an elliptic linear differential operator of order $m$ with ${\mathcal{C}}^{\infty }$ coefficients operating on the sections of a complex vector bundle $\displaystyle G:=(H,\pi ,M)$ over a compact Riemannian manifold $M$ without boundary and $\omega \in L^{r}_{G}(M)\cap (\mathrm{k}\mathrm{e}\mathrm{r}D^{*})^{\perp },$ then there is a $u\in W^{m,r}_{G}(M)$ such that $Du=\omega $ on $M.$ \quad Next we investigate the case of a compact manifold with boundary by use of the "riemannian double manifold". In the last sections we study the more delicate case of a complete but non compact Riemannian manifold by use of adapted weights.

The main result of this Ph.D. thesis is to present conditions on the curvature and boundary of an orientable, compact manifold under which there is a unique global solution to the Dirichlet boundary value problem (BVP) for the prescribed Ricci curvature equation. This Dirichlet BVP is a determined, non-elliptic system of 2nd-order, quasilinear partial differential equations, which is supplemented with a constraint equation in the form of the so-called Bianchi identity. Indeed, in order to prove the main result of this Ph.D. thesis, we are also required to prove that the kernel of the Bianchi operator is a smooth tame Frechet submanifold of the space of Riemannian metrics on a compact Riemannian manifold with boundary. After presenting an overview of the literature in Chapter 1 and the required notation and background in Riemannian geometry and geometric analysis in Chapter 2, the main results of this thesis are organised into the following two chapters.In Chapter 3 we present conditions under which the kernel of the Bianchi operator is globally a smooth tame Frechet submanifold of the space of Riemannian metrics on a Riemannian compact manifold both with or without boundary. This global submanifold result for the kernel of the Bianchi operator is central to the proof of global existence and uniqueness for the Dirichlet BVP for the Ricci curvature equation, and extends the analogous local submanifold result by Dennis DeTurck in [18] which was fundamental to the proof in [18] of local existence and uniqueness of the prescribed Ricci curvature equation on a compact manifold without boundary.Furthermore, the material in Chapter 3 of this thesis sits more generally within the literature on linearisation stability of nonlinear PDE. Indeed, the method by which we prove that the kernel of the Bianchi operator is a global submanifold of the space of metrics on a compact manifold is an application of the more general technique of proving the linearisation stability of a system of nonlinear PDE, and yields that the Bianchi operator is itself linearisation stable in a sense made precise in Chapter 3.In Chapter 4 we then use the fact that the kernel of the Bianchi operator is globally a smooth tame Frechet manifold, and the Nash-Moser implicit function theorem (IFT) in the smooth tame category, to find and conditions under which globally there is a unique Riemannian metric satisfying the Dirichlet BVP for the Ricci curvature equation. This global existence and uniqueness result is motivated by, and can be viewed as a modification of, an analogous result for Einstein manifolds of negative sectional curvature and convex and umbilical boundary presented by Jean-Marc Schlenker in [50]. Loosely speaking, in the context of Einstein manifolds the kernel of the Bianchi operator is the entire space of Riemannian metrics and thus automatically a smooth tame Frechet manifold; however, in the context of the prescribed Ricci curvature equation, in which this Ph.D. thesis is interested, we are required to prove that the kernel is a smooth tame Frechet submanifold in order to apply the Nash-Moser IFT.In general, global existence for the prescribed Ricci curvature equation on a manifold with or without boundary is difficult and results in this direction are few; the main goal of this thesis makes an important contribution in this area and the techniques used to prove it have potential applications in other areas of geometric analysis such as Yang-Mills gauge theory and curvature flows.