Taoufik Bouziane
In this paper, we prove the existence of energy minimizers in each free homotopy class of maps between polyhedra with target space without focal points. Our proof involves a careful study of some geometric properties of riemannian polhyedra without focal points. Among other things, we show that on the relevant polyhedra, there exists a convex supporting function.
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The goal of the present paper is to establish some kind of regularity of an energy minimizer map between Riemannian polyhedra. More precisely, we will show the hölder continuity of local energy minimizers between Riemannian polyhedra with the target spaces without focal points. With this new result, we also complete our existence theorem obtained in [5], and consequently we generalize completely, to the case of target polyhedra without focal points (which is weaker geometric condition than the nonpositivity of the curvature), the Eells-fuglede's existence and regularity theorem [12, chapters 1…
Approximating Riemannian manifolds by polyhedra
本文在受限黎曼框架下研究用多面体逼近黎曼流形，并沿 Regge 的原始直观思路给出 Regge 定理的证明，说明标量曲率积分可由多面体曲率逼近。
We give an account of some recent development that connects the concept of mass in general relativity to the geometry of large Riemannian polyhedra, in the setting of both asymptotically flat and asymptotically hyperbolic manifolds.
We show that the concept of the ADM mass in general relativity can be understood as the limit of the total mean curvature plus the total defect of dihedral angle of the boundary of large Riemannian polyhedra. We also express the $n$-dimensional mass as a suitable integral of geometric quantities that determine the $(n-1)$-dimensional mass.
We prove the $C^{1}$ regularity for a class of abnormal length-minimizers in rank $2$ sub-Riemannian structures. As a consequence of our result, all length-minimizers for rank $2$ sub-Riemannian structures of step up to $4$ are of class $C^{1}$.
We state that any constant curvature Riemannian metric with conical singularities of constant sign curvature on a compact (orientable) surface $S$ can be realized as a convex polyhedron in a Riemannian or Lorentzian) space-form. Moreover such a polyhedron is unique, up to global isometries, among convex polyhedra invariant under isometries acting on a totally umbilical surface. This general statement falls apart into 10 different cases. The cases when $S$ is the sphere are classical.
On Riemannian polyhedra with non-obtuse dihedral angles in 3-manifolds with positive scalar curvature
本文确定了R^3中三维简单凸多面体的组合类型，其能在正数量曲率黎曼3-流形中实现为具非钝二面角的黎曼多面体，并视为Andreev定理的类比，同时构造了高维例子。
This paper studies a variational obstacle avoidance problem on complete Riemannian manifolds. That is, we minimize an action functional, among a set of admissible curves, which depends on an artificial potential function used to avoid obstacles. In particular, we generalize the theory of bi-Jacobi fields and biconjugate points and present necessary and sufficient conditions for optimality. Local minimizers of the action functional are divided into two categories and subsequently classified, with local uniqueness results obtained in both cases.
We have completely rewritten the paper, and corrected the proofs. We construct an exponential map at any point in the (n-1)-skeleton minus the (n-2)-skeleton of an n-dimensional Riemannian polyhedron. We have added allover the extra-assumption that the exponential map is totally geodesic at points in the (n-1)-skeleton minus the (n-2)-skeleton. This condition is automatically realized for instance if the dimension is one, or if the metric is flat.
The aim of this paper is to extend the notion of pseudo harmonic morphism (introduced by Loubeau \cite {Lo}) to the case when the source manifold is an admissible Riemannian polyhedron. We define these maps to be harmonic in the sense of Eells-Fuglede \cite {EF} and pseudo-horizontally weakly conformal in our sense (see Section 3). We characterize them by means of germs of harmonic functions on the source polyhedron, in sense of Korevaar-Schoen \cite {KS}, and germs of holomorphic functions on the Kähler target manifold.
We consider the nilpotent left-invariant sub-Riemannian structure on the Engel group. This structure gives a fundamental local approximation of a generic rank 2 sub-Riemannian structure on a 4-manifold near a generic point (in particular, of the kinematic models of a car with a trailer). On the other hand, this is the simplest sub-Riemannian structure of step three. We describe the global structure of the cut locus (the set of points where geodesics lose their global optimality), the Maxwell set (the set of points that admit more than one minimizer), and the intersection of the cut locus with …
We show the existence of a smooth spherical surface minimizing the Willmore functional subject to an area constraint in a compact Riemannian three-manifold, provided the area is small enough. Moreover, we classify complete surfaces of Willmore type with positive mean curvature in Riemannian three-manifolds.
Not all sub-Riemannian minimizing geodesics are smooth
本文否定了次黎曼长度极小化子必然光滑的猜想，通过构造一个实解析（甚至多项式）次黎曼结构，证明了存在属于 $C^2$ 但非 $C^3$ 的长度极小化子。
We prove nonexistence of nonconstant local minimizers for a class of functionals, which typically appears in the scalar two-phase field model, over a smooth N-dimensional Riemannian manifold without boundary with non-negative Ricci curvature. Conversely for a class of surfaces possessing a simple closed geodesic along which the Gauss curvature is negative we prove existence of nonconstant local minimizers for the same class of functionals.
We determine regularity results for energy minimizing maps from an $n$-dimensional Riemannian polyhedral complex $X$ into a CAT(1) space. Provided that the metric on $X$ is Lipschitz regular, we prove Hölder regularity with Hölder constant and exponent dependent on the total energy of the map and the metric on the domain. Moreover, at points away from the $(n-2)$-skeleton, we improve the regularity to locally Lipschitz. Finally, for points $x \in X^{(k)}$ with $k \leq n-2$, we demonstrate that the Hölder exponent depends on geometric and combinatorial data of the link of $x \in X$.
In the main theorem of this paper we treat the problem of existence of minimizers of the isoperimetric problem under the assumption of small volumes. Applications of the main theorem to asymptotic expansions of the isoperimetric problem are given.
We let (M^m, g) be a closed smooth Riemannian manifold (m >1) with positive scalar curvature S_g, and prove that the Yamabe constant of (M \times R^n,g+g_E) is achieved by a metric in the conformal class of (g+g_E), where g_E is the Euclidean metric. We also show that the Yamabe quotient of (M \times R^n,g+g_E) is improved by Steiner symmetrization with respect to M. It follows from this last assertion that the dependence on R^n of the Yamabe minimizer of (M \times R^n,g+g_E) is radial.
In this paper, we present strongly geodesic preinvexity on Riemannian manifolds (RM) and strongly η-invexity of order m on RM. Furthermore, we define strongly invariant η-monotonicity of order m on RM. Under Condition C, an important characterization of these functions are studied. We construct several non-trivial examples in support of these definitions. Afterwords, an important and significant characterization of a strict η-minimizers (η-minimizers)of order m for MOP and a solution to the variational like-inequality problem (VVLIP) has been derived.
We consider the problem of minimizing $\int_{0}^L \sqrt{ξ^2 +K^2(s)}\, ds $ for a planar curve having fixed initial and final positions and directions. The total length $L$ is free. Here $s$ is the variable of arclength parametrization, $K(s)$ is the curvature of the curve and $ξ>0$ a parameter. This problem comes from a model of geometry of vision due to Petitot, Citti and Sarti. We study existence of local and global minimizers for this problem. We prove that if for a certain choice of boundary conditions there is no global minimizer, then there is neither a local minimizer nor a geodesic. W…
For a Riemannian polyhedra, we study the geometry of the unit ball for the unidimensional stable norm (stable ball). In the case of a unidimensional Riemannian polyhedra (graph), we show that the stable ball is a polytope whose vertices are completely described by combinatorial properties of the graph. We study then the realizable forms as stable ball of Riemannan manifolds of dimension larger than three. For a Riemannian manifold $(M, g)$ fixed, we show that a broad class of polytopes can appear as stable ball of metrics in the conformal class of $g$. We use for that a polyhedral technique.
In this note we discuss the geometry of Riemannian surfaces having a discrete set of singular points. We assume the conformal structure extends through the singularities and the curvature is integrable. Such points are called \emph{simple singularities}. We first describe them locally and then globally using the notion of (real) divisor. We formulate a Gauss-Bonnet formula and relate it to some asymptotic isoperimetric ratio. We prove a classifications theorem for flat metrics with simple singularities on a compact surface and discuss the Berger--Nirenberg Problem on surfaces with a divisor. W…
We study analytic properties of harmonic maps from Riemannian polyhedra into CAT($κ$) spaces for $κ\in\{0,1\}$. Locally, on each top-dimensional face of the domain, this amounts to studying harmonic maps from smooth domains into CAT($κ$) spaces. We compute a target variation formula that captures the curvature bound in the target, and use it to prove an $L^p$ Liouville-type theorem for harmonic maps from admissible polyhedra into convex CAT($κ$) spaces. Another consequence we derive from the target variation formula is the Eells-Sampson Bochner formula for CAT(1) targets.
We prove existence and partial regularity of integral rectifiable $m$-dimensional varifolds minimizing functionals of the type $\int |H|^p$ and $\int |A|^p$ in a given Riemannian $n$-dimensional manifold $(N,g)$, $2\leq m<n$ and $p>m$, under suitable assumptions on $N$ (in the end of the paper we give many examples of such ambient manifolds). To this aim we introduce the following new tools: some monotonicity formulas for varifolds in $\mathbb{R}^S$ involving $\int |H|^p$, to avoid degeneracy of the minimizer, and a sort of isoperimetric inequality to bound the mass in terms of the mentioned f…
Submanifolds of Constant Negative Curvature: A Generalization of Hilbert's Theorem
本文通过广义高斯-博内公式证明n维双曲空间无法嵌入到2n-1维欧几里得空间，推广了希尔伯特定理。
A Variational Approach to the Yamabe Problem: Conformal Transformations and Scalar Curvature on Compact Riemannian Manifolds
本文用变分与集中紧性方法研究紧Riemann流形上的Yamabe问题，证明在$\lambda(M)<\lambda(S^n)$条件下存在共形常标量曲率度量。
Consider a closed Riemannian $n$-manifold $M$ admitting a negatively curved Riemannian metric. We show that for every Riemannian metric on $M$ of sufficiently small volume, there is a point in the universal cover of $M$ such that the volume of every ball of radius $r \geq 1$ centered at this point is greater or equal to the volume of the ball of the same radius in the hyperbolic $n$-space. We also give an interpretation of this result in terms of macroscopic scalar curvature. This result, which holds more generally in the context of polyhedral length spaces, is related to a question of Guth. I…
Given a compact Riemannian manifold $(M^n,g)$ and a fixed cohomology class, $[α^*] \in H^k(M)$, we consider the existence of a minimizer $α\in [α^*]$ of the generalized minimal surface energy $\int_M \sqrt{1+|α|^2} dV_g$. When $k = 1$, we prove the existence of unique minimizers for every cohomology class $[α^*]$. Next, when $k > 1$, we construct examples of singular solutions for finite cohomology class $[α^*] \in H^k(S^k \times S^k,g)$, where $g$ is conformal to the standard metric on $S^k \times S^k$. Additionally, we show that when $k=2$, these singular solutions are also solutions to the …
A note on the regularity and the existence of Riemannian splines
本文利用 DuBois-Reymond 引理的推广，补全了黎曼流形上高阶样条能量泛函临界点正则性的详细证明，并证明了在给定多插值点和单一速度条件下极小值子的存在性。
Given two-dimensional Riemannian manifolds $\mathcal{M},\mathcal{N}$, we prove a lower bound on the distortion of embeddings $\mathcal{M} \to \mathcal{N}$, in terms of the areas' discrepancy $V_{\mathcal{N}}/V_{\mathcal{M}}$, for a certain class of distortion functionals. For $V_{\mathcal{N}}/V_{\mathcal{M}} \ge 1/4$, homotheties, provided they exist, are the unique energy minimizing maps attaining the bound, while for $V_{\mathcal{N}}/V_{\mathcal{M}} \le 1/4$, there are non-homothetic minimizers. We characterize the maps attaining the bound, and construct explicit non-homothetic minimizers be…
The paper is devoted to the variational analysis of the Willmore, and other L^2 curvature functionals, among immersions of 2-dimensional surfaces into a compact riemannian m-manifold (M^m,h) with m>2. The goal of the paper is twofold, on one hand, we give the right setting for doing the calculus of variations (including min max methods) of such functionals for immersions into manifolds and, on the other hand, we prove existence results for possibly branched Willmore spheres under various constraints (prescribed homotopy class, prescribed area) or under curvature assumptions for M^m. To this ai…
For a sub-Riemannian manifold provided with a smooth volume, we relate the small time asymptotics of the heat kernel at a point $y$ of the cut locus from $x$ with roughly "how much" $y$ is conjugate to $x$. This is done under the hypothesis that all minimizers connecting $x$ to $y$ are strongly normal, i.e.\ all pieces of the trajectory are not abnormal. Our result is a refinement of the one of Leandre $4t\log p_t(x,y)\to -d^2(x,y)$ for $t\to 0$, in which only the leading exponential term is detected. Our results are obtained by extending an idea of Molchanov from the Riemannian to the sub-Rie…
We investigate rigidity and stability properties of critical points of quadratic curvature functionals on the space of Riemannian metrics. We show it is possible to "gauge" the Euler-Lagrange equations, in a self-adjoint fashion, to become elliptic. Fredholm theory may then be used to describe local properties of the moduli space of critical metrics. We show a number of compact examples are infinitesimally rigid, and consequently, are isolated critical points in the space of unit-volume Riemannian metrics. We then give examples of critical metrics which are strict local minimizers (up to diffe…
The Bernstein problem in higher dimensional Heisenberg groups
针对高维Heisenberg群中的Bernstein问题，证明存在边界为光滑非特征整体内禀图但非垂直超平面的全局周长极小化子。
Given a Riemannian 3-ball $(\bar B, g)$ of non-negative scalar curvature, Bartnik conjectured that $(\bar B, g)$ admits an asymptotically flat (AF) extension (without horizons) of the least possible ADM mass, and that such a mass-minimizer is an AF solution to the static vacuum Einstein equations, uniquely determined by natural geometric conditions on the boundary data of $(\bar B, g)$. We prove the validity of the second statement, i.e.~such mass-minimizers, if they exist, are indeed AF solutions of the static vacuum equations. On the other hand, we prove that the first statement is not true …
A morph between two Riemannian $n$-manifolds is an isotopy between them together with the set of all intermediate manifolds equipped with Riemannian metrics. We propose measures of the distortion produced by some classes of morphs and diffeomorphisms between two isotopic Riemannian $n$-manifolds and, with respect to these classes, prove the existence of minimal distortion morphs and diffeomorphisms. In particular, we consider the class of time-dependent vector fields (on an open subset $Ω$ of $ \R^{n+1}$ in which the manifolds are embedded) that generate morphs between two manifolds $M$ and $N…
The variational problem for the functional $F=\frac12\|ϕ^*ω\|_{L^2}^2$ is considered, where $ϕ:(M,g)\to (N,ω)$ maps a Riemannian manifold to a symplectic manifold. This functional arises in theoretical physics as the strong coupling limit of the Faddeev-Hopf energy, and may be regarded as a symplectic analogue of the Dirichlet energy familiar from harmonic map theory. The Hopf fibration $π:S^3\to S^2$ is known to be a locally stable critical point of $F$. It is proved here that $π$ in fact minimizes $F$ in its homotopy class and this result is extended to the case where $S^3$ is given the metr…
We prove partial regularity of stationary solutions and minimizers $u$ from a set $Ω\subset \mathbb R^n$ to a Riemannian manifold $N$, for the functional $\int_ΩF(x,u,|\nabla u|^2) dx$. The integrand $F$ is convex and satisfies some ellipticity and boundedness assumptions. We also develop a new monotonicity formula and an $ε$-regularity theorem for such stationary solutions with no restriction on their images. We then use the idea of quantitative stratification to show that the k-th strata of the singular set of such solutions are k-rectifiable.
We study the problem of finding the global Riemannian center of mass of a set of data points on a Riemannian manifold. Specifically, we investigate the convergence of constant step-size gradient descent algorithms for solving this problem. The challenge is that often the underlying cost function is neither globally differentiable nor convex, and despite this one would like to have guaranteed convergence to the global minimizer. After some necessary preparations we state a conjecture which we argue is the best (in a sense described) convergence condition one can hope for. The conjecture specifi…
Uniform, localized asymptotics for sub-Riemannian heat kernels, their logarithmic derivatives, and associated diffusion bridges
本研究证明次黎曼热核及其导数、对数导数的小时间渐近性可局部定位，建立了其与极小测地线结构的关联，得到紧集一致界与展开式，还刻画了非异常割迹的特征。
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