In this paper a compact Riemannian manifold with strictly convex boundary is reconstructed from its partial travel time data. This data assumes that an open measurement region on the boundary is given, and that for every point in the manifold, the respective distance function to the points on the measurement region is known. This geometric inverse problem has many connections to seismology, in particular to micro seismicity. The reconstruction is based on embedding the manifold in a function space. This requires the differentiation of the distance functions. Therefore this paper also studies some global regularity properties of the distance function on a compact Riemannian manifold with strictly convex boundary.
We provide new proofs based on the Myers-Steenrod theorem to confirm that travel time data, travel time difference data and the broken scattering relations determine a simple Riemannian metric on a disc up to the natural gauge of a boundary fixing diffeomorphism. Our method of the proof leads to a Lipschitz-type stability estimate for the first two data sets in the class of simple metrics.
We show that the travel time difference functions, measured on the boundary, determine a compact Riemannian manifold with smooth boundary up to Riemannian isometry, if boundary satisfies a certain visibility condition. This corresponds with the inverse microseismicity problem. The novelty of our paper is a new type of a proof and a weaker assumption for the boundary than it has been presented in the literature before. We also construct an explicit smooth atlas from the travel time difference functions.
Using recent work of Bettiol, we show that a first-order conformal deformation of Wilking's metric of almost-positive sectional curvature on $S^2\times S^3$ yields a family of metrics with strictly positive average of sectional curvatures of any pair of 2-planes that are separated by a minimal distance in the 2-Grassmanian. A result of Smale's allows us to conclude that every closed simply connected 5-manifold with torsion-free homology and trivial second Stiefel-Whitney class admits a Riemannian metric with a strictly positive average of sectional curvatures of any pair of orthogonal 2-planes…
We propose a new approach to the study of compact Riemannian manifolds with nonnegative Ricci curvature and strictly convex boundary or positive Ricci curvature and convex boundary. Several conjectures are formulated. Some partial results that support these conjectures are established.
Let $M$ be a complete Riemannian manifold possessing a strictly convex Lipschitz continuous exhaustion function. We show that the isoperimetric profile of $M$ is a continuous and non-decreasing function. Particular cases are Hadamard manifolds and complete non-compact manifolds with strictly positive sectional curvatures.
In a complete simply connected Riemannian manifold X of pinched negative curvature, we give a sharp criterion for a subset C to be the epsilon-neighbourhood of some convex subset of X, in terms of the extrinsic curvatures of the boundary of C.
In this paper, we give some rigidity results for both harmonic and pseudoharmonic maps from CR manifolds into Riemannian manifolds or Kahler manifolds. Some basicity, pluriharmonicity and Siu-Sampson type results are established for both harmonic maps and pseudoharmonic maps.
We prove that the existence of one flat horosphere in the universal cover of a closed, strictly quarter pinched, negatively curved Riemannian manifold of dimension n with n greater than or equal to 3, implies that the manifold is homothetic to a real hyperbolic manifold.
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). For example, if a convex set in (M,g) is bounded by a smooth hypersurface, then it is strictly convex.
In this paper we have proved that a compact Riemannian manifold does not admit a metric with positive scalar curvature if there exists a real valued function in this manifold which is strictly positive along a geodesic ray satisfying expanding or steady Yamabe soliton. We have also deduced a relation between scalar curvature and surface area of a geodesic ball in a Riemannian manifold with a pole satisfying steady Yamabe soliton.
We prove curvature estimates for general curvature functions. As an application we show the existence of closed, strictly convex hypersurfaces with prescribed curvature $F$, where the defining cone of $F$ is $\C_+$. $F$ is only assumed to be monotone, symmetric, homogeneous of degree 1, concave and of class $C^{m,\al}$, $m\ge4$.
We characterize certain CR structures of arbitrary codimension (different from 3, 4 and 5) on Riemannian Spin$^c$ manifolds by the existence of a Spin$^c$ structure carrying a strictly partially pure spinor field. Furthermore, we study the geometry of Riemannian Spin$^c$ manifolds carrying a strictly partially pure spinor which satisfies the generalized Killing equation in prescribed directions.
Given a smooth non-trapping compact manifold with strictly con- vex boundary, we consider an inverse problem of reconstructing the manifold from the scattering data initiated from internal sources. This data consist of the exit directions of geodesics that are emaneted from interior points of the manifold. We show that under certain generic assumption of the metric, one can reconstruct an isometric copy of the manifold from such scattering data measured on the boundary.
For a closed Riemannian manifold $(M^m,g)$ of constant positive scalar curvature and any other closed Riemannian manifold $(N^n,h)$, we show that the limit of the Yamabe constants of the Riemannian products $(M\times N,g+rh)$ as $r$ goes to infinity is equal to the Yamabe constant of $(M^m \times R^n, [g+g_E])$ and is strictly less than the Yamabe invariant of $S^{m+n}$ provided $n\geq 2$. We then consider the minimum of the Yamabe functional restricted to functions of the second variable and we compute the limit in terms of the best constants of the Gagliardo-Nirenberg inequalities.
In view of A. Andreotti and H. Grauert's vanishing theorem for q-complete domains in C^n, (Théorème de finitude pour la cohomologie des espaces complexes, Bull. Soc. Math. France 90 (1962), 193--259,) we re-prove a vanishing result by J.-P. Sha, (p-convex Riemannian manifolds, Invent. Math. 83 (1986), no. 3, 437--447,) and H. Wu, (Manifolds of partially positive curvature, Indiana Univ. Math. J. 36 (1987), no. 3, 525--548,) for the de Rham cohomology of strictly p-convex domains in R^n in the sense of F. R. Harvey and H. B. Lawson, (The foundations of p-convexity and p-plurisubharmonicity in r…
We introduce the notion of translational Riemannian manifolds and define a Gauss map for orientable immersed hypersurfaces lying in these ambients, an associated translational curvature and prove a Gauss-Bonnet theorem. We also use this Gauss map to prove that if $M^{n}$ is a compact, connected and oriented immersed hypersurface of the unit sphere $\mathbb{S}^{n+1}$ ($n\geq2$) contained in a geodesic ball of radius $R$ and whose principal curvatures are strictly bigger than $\tan\left( R/2 \right)$, then $M$ is diffeomorphic to $\mathbb{S}^{n}$. Additionally, we show that for any $\varepsilon\…
For a subRiemannian manifold and a given Riemannian extension of the metric, we define a canonical global connection. This connection coincides with both the Levi-Civita connection on Riemannian manifolds and the Tanaka-Webster connection on strictly pseudoconvex CR manifolds. We define a notion of normality generalizing Tanaka's notion for CR manifolds to the subRiemannian case. Under the assumption of normality, we construct local frames that simplify computations in a manner analogous to Riemannian normal coordinates. We then use these frames to establish Bianchi Identities and symmetries f…
The notion of strictly outward minimising hull is investigated for open sets of finite perimeter sitting inside a complete noncompact Riemannian manifold. Under natural geometric assumptions on the ambient manifold, the strictly outward minimising hull $Ω^*$ of a set $Ω$ is characterised as a maximal volume solution of the least area problem with obstacle, where the obstacle is the set itself. In the case where $Ω$ has $\mathscr{C}^{1, α}$-boundary, the area of $\partial Ω^*$ is recovered as the limit of the $p$-capacities of $Ω$, as $p \to 1^+$. Finally, building on the existence of strictly …
We prove that any smooth Riemannian manifold of non-negative scalar curvature and with a strictly mean convex and compact boundary component can be (C^2) extended beyond the component to have non-negative scalar curvature and to enjoy anyone of the following three types of (new) boundary: strictly convex, totally geodesic or strictly concave. The extension procedure can be applied for instance to "positive mass" type of theorems.
Let $C$ be a strictly convex domain in a $3$-dimensional Riemannian manifold with sectional curvature bounded above by a constant and let $Σ$ be a constant mean curvature surface with free boundary in $C$. We provide a pinching condition on the length of the traceless second fundamental form on $Σ$ which guarantees that the surface is homeomorphic to either a disk or an annulus. Furthermore, under the same pinching condition, we prove that if $C$ is a geodesic ball of $3$-dimensional space forms, then $Σ$ is either a spherical cap or a Delaunay surface.
We analyze the inverse problem, originally formulated by Dix in geophysics, of reconstructing the wave speed inside a domain from boundary measurements associated with the single scattering of seismic waves. We consider a domain $\tilde M$ with a varying and possibly anisotropic wave speed which we model as a Riemannian metric $g$. For our data, we assume that $\tilde M$ contains a dense set of point scatterers and that in a subset $U\subset \tilde M$, modeling a region containing measurement devices, we can measure the wave fronts of the single scattered waves diffracted from the point scatte…
We establish a second order smooth variational principle valid for functions defined on (possibly infinite-dimensional) Riemannian manifolds which are uniformly locally convex and have a strictly positive injectivity radius and bounded sectional curvature.
We consider the graphical mean curvature flow of strictly area decreasing maps $f:M\to N$, where $M$ is a compact Riemannian manifold of dimension $m>1$ and $N$ a complete Riemannian surface of bounded geometry. We prove long-time existence of the flow and that the strictly area decreasing property is preserved, when the bi-Ricci curvature $BRic_M$ of $M$ is bounded from below by the sectional curvature $σ_N$ of $N$. In addition, we obtain smooth convergence to a minimal map if $Ric_M\ge\sup\{0,{\sup}_Nσ_N\}$. These results significantly improve known results on the graphical mean curvature fl…
Riemannian manifolds with non-zero Killing spinors are Einstein manifolds. Klaus Kröncke proved that all complete Riemannian manifolds with imaginary Killing spinors are (linearly) strictly stable in \cite{Kro15}. In this paper, we obtain a new proof for this stability result by using a Bochner type formula in \cite{DWW05} and \cite{Wan91}. Moreover, existence of real Killing spinors is closely related to the Sasaki-Einstein structure. A regular Sasaki-Einstein manifold is essentially the total space of a certain principal $S^{1}$-bundle over a Kähler-Einstein manifold. We prove that if the ba…
We investigate stability and local minimizing properties of the Riemannian functional defined by the L^p norm of the curvature tensor on the space of Riemannian metrics on a closed manifold. Riemannian metrics with constant curvature and products of such metrics are critical points of this functional. We prove that these points are strictly stable for this functional and if (M; g) is a manifold of this type, g has a neighborhood U such that g is the strict minima on it.
Let $\,(M,g)\,$ be a $n$-dimensional Riemannian manifold and $\,Ω\,$ be any compact connected domain in $\,M$. We study the problem of finding the {\em maxima} of the functional $\, {\mathcal E} (Ω)\,$ (known as {\em torsional rigidity} associated to $Ω$) among all domains of prescribed volume $v$. Our results show that for a given Riemannian manifold which is strictly isoperimetric at one of its points the maximum of such functional is realized by the geodesic ball centered at this point. More generally, we prove estimates for the functional $\, {\mathcal E} (Ω)\,$ by comparison with symmetri…
In this paper, we consider the high order geometric flows of a submanifolds $M$ in a complete Riemannian manifold $N$ with $\dim(N)=\dim(M)+1=n+1$, which were introduced by Mantegazza in the case the ambient space is an Euclidean space, and extend some results due to Mantegazza to the present situation under some assumptions on $N$. Precisely, we show that if $m\in\mathbb{N}$ is strictly larger than the integer part of $n/2$ and $φ(t)$ is a immersion for all $t\in[0,T)$ and if $\mathfrak{F}_m(φ_0)$ is bounded by a constant which relies on the injectivity radius $\bar{R}>0$ and sectional curvat…
Interior singularity and branching of geodesics in real-analytic sub-Riemannian manifolds
Università degli Studi dell’Aquila（阿奎拉大学）；Scuola Internazionale Superiore di Studi Avanzati (SISSA)（国际高等研究院）；Unité de Mathématiques Appliquées, ENSTA, Institut Polytechnique de Paris（巴黎理工学院应用数学系）
Let M be a compact manifold of dimension n with a strictly convex projective structure. We consider the geodesic flow of the Hilbert metric on it, which is known to be Anosov. We prove that its topological entropy is less than n-1, with equality if and only if the structure is Riemannian, that is hyperbolic. As a corollary, we get that the volume entropy of a divisible strictly convex set is less than n-1, with equality if and only if it is an ellipsoid.
We prove some Liouville type theorems on smooth compact Riemannian manifolds with nonnegative sectional curvature and strictly convex boundary. This gives a nonlinear generalization in low dimension of the recent sharp lower bound of the first Steklov eigenvalue by Xia-Xiong and verifies partially a conjecture by the third author. As a consequence, we derive several sharp Sobolev trace inequalities on these manifolds.
It was proved in \cite{NS1} that obstacles $K$ in $\R^d$ that are finite disjoint unions of strictly convex domains with $C^3$ boundaries are uniquely determined by the travelling times of billiard trajectories in their exteriors and also by their so called scattering length spectra. However the case $d = 2$ is not properly covered in \cite{NS1}. In the present paper we give a separate different proof of the same result in the case $d = 2$.
We survey some results on travel time tomography. The question is whether we can determine the anisotropic index of refraction of a medium by measuring the travel times of waves going through the medium. This can be recast as geometry problems, the boundary rigidity problem and the lens rigidity problem. The boundary rigidity problem is whether we can determine a Riemannian metric of a compact Riemannian manifold with boundary by measuring the distance function between boundary points. The lens rigidity problem problem is to determine a Riemannian metric of a Riemannian manifold with boundary …
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 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 same marked boundary distance, extending a result of Croke and Otal.
We obtain Harnack estimates for a class of curvature flows in Riemannian manifolds of constant non-negative sectional curvature as well as in the Lorentzian Minkowski and de Sitter spaces. Furthermore, we prove a Harnack estimate with a bonus term for mean curvature flow in locally symmetric Riemannian Einstein manifold of non-negative sectional curvature. Using a concept of "duality" for strictly convex hypersurfaces, we also obtain a new type of inequalities, so-called "pseudo"-Harnack inequalities, for expanding flows in the sphere and in the hyperbolic space.
For compact Riemannian manifolds with convex boundary, B.White proved the following alternative: Either there is an isoperimetric inequality for minimal hypersurfaces or there exists a closed minimal hypersurface, possibly with a small singular set. There is the natural question if a similar result is true for submanifolds of higher codimension. Specifically, B.White asked if the non-existence of an isoperimetric inequality for k-varifolds implies the existence of a nonzero, stationary, integral k-varifold. We present examples showing that this is not true in codimension greater than two. The …
A stochastic flow is constructed on a frame bundle adapted to a Riemannian foliation on a compact manifold. The generator A of the resulting transition semigroup is shown to preserve the basic functions and forms, and there is an essentially unique strictly positive smooth function phi satisfying A^* phi = 0. This function is used to perturb the metric, and an application of the ergodic theorem shows that there exists a bundle-like metric for which the basic projection of the mean curvature is basic-harmonic.
Suppose the Riemannian metrics $g$ and $\bar g$ on a closed connected manifold $M^n$ are geodesically equivalent and strictly non-proportional at least at one point. Then the topological entropy of the geodesic flow of $g$ vanishes.
Having in mind the well known model of Euclidean convex hypersurfaces [4], [5], and the ideas in [1] many authors defined and investigate convex hypersurfaces of a Riemannian manifold. As it was proved by the first author in [7], there follows the interdependence between convexity and Gauss curvature of the hypersurface. In this paper we define $H$-convexity of a Riemannian submanifold of arbitrary codimension, replacing the normal versor of a hypersurface with the mean curvature vector $H$. A characterization, used by B.Y. Chen [2], [3] as the definition of strictly $H$-convexity, it is obtai…
Let $M^n$ be either a simply connected space form or a rank-one symmetric space of noncompact type. We consider Weingarten hypersurfaces of $M\times\mathbb R$, which are those whose principal curvatures $k_1,\dots ,k_n$ and angle function $θ$ satisfy a relation $W(k_1,\dots,k_n,θ^2)=0,$ being $W$ a differentiable function which is symmetric with respect to $k_1,\dots, k_n.$ When $\partial W/\partial k_i>0$ on the positive cone of $\mathbb R^n,$ a strictly convex Weingarten hypersurface determined by $W$ is said to be elliptic. We show that, for a certain class of Weingarten functions $W,$ ther…
