Compact Riemannian Manifold Of Dimension Research Articles

Dimension estimates and approximation in non-uniformly hyperbolic systems

Abstract Let $f: M\rightarrow M$ be a $C^{1+\alpha }$ diffeomorphism on an $m_0$ -dimensional compact smooth Riemannian manifold M and $\mu $ a hyperbolic ergodic f-invariant probability measure. This paper obtains an upper bound for the stable (unstable) pointwise dimension of $\mu $ , which is given by the unique solution of an equation involving the sub-additive measure-theoretic pressure. If $\mu $ is a Sinai–Ruelle–Bowen (SRB) measure, then the Kaplan–Yorke conjecture is true under some additional conditions and the Lyapunov dimension of $\mu $ can be approximated gradually by the Hausdorff dimension of a sequence of hyperbolic sets $\{\Lambda _n\}_{n\geq 1}$ . The limit behaviour of the Carathéodory singular dimension of $\Lambda _n$ on the unstable manifold with respect to the super-additive singular valued potential is also studied.

Multiplicity results for nonhomogenous elliptic equation involving the generalized Paneitz-Branson operator

"Let (M, g) be a compact Riemannian manifold of dimension n ≥ 3 without boundary ∂M, we consider the multiplicity result of solutions of the following nonhomogenous fourth order elliptic equation involving the generalized Paneitz-Branson operator, Pg (u) = f (x) |u|2 −2 u + h(x). Under some conditions and using critical points theory, we prove the existence of two distinct solutions of the above equation. At the end, we give a geometric example when the equation has negative and positive solutions. Keywords: Riemannian manifold, multiplicity result, nonhomogeneous, Paneitz-Branson operator, critical points theory."

Graphical mean curvature flow with bounded bi-Ricci 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 $\sigma_N$ of $N$. In addition, we obtain smooth convergence to a minimal map if $Ric_M\ge\sup\{0,{\sup}_N\sigma_N\}$. These results significantly improve known results on the graphical mean curvature flow in codimension $2$.

Structural descriptions of limits of the parabolic Ginzburg-Landau equation on closed manifolds

In the setting of a compact Riemannian manifold of dimension $N\ge3$, we provide a structural description of the limiting behavior of the energy measures of solutions to the parabolic Ginzburg-Landau equation. In particular, we provide a decomposition of the limiting energy measure into a diffuse part, which is absolutely continuous with respect to the volume measure, and a concentrated part supported on a codimension $2$ rectifiable subset. We also demonstrate that the time evolution of the diffuse part is determined by the heat equation while the concentrated part evolves according to a Brakke flow. This paper extends the work of Bethuel, Orlandi, and Smets from [8].

Generic existence of multiplicity-1 minmax minimal hypersurfaces via Allen–Cahn

In Guaraco (J. Differential Geom. 108(1):91–133, 2018) a new proof was given of the existence of a closed minimal hypersurface in a compact Riemannian manifold N^{n+1} with nge 2. This was achieved by employing an Allen–Cahn approximation scheme and a one-parameter minmax for the Allen–Cahn energy (relying on works by Hutchinson, Tonegawa, Wickramasekera to pass to the limit as the Allen-Cahn parameter tends to 0). The minimal hypersurface obtained may a priori carry a locally constant integer multiplicity. Here we modify the minmax construction of Guaraco (J. Differential Geom. 108(1):91–133, 2018), by allowing an initial freedom on the choice of the valley points between which the mountain pass construction is carried out, and then optimising over said choice. We then prove that, when 2le nle 6 and the metric is bumpy, this minmax leads to a (smooth closed) minimal hypersurface with multiplicity 1. (When n=2 this conclusion also follows from Chodosh and Mantoulidis (Ann. Math. 191(1):213–328, 2020).) As immediate corollary we obtain that every compact Riemannian manifold of dimension n+1, 2le nle 6, endowed with a bumpy metric, admits a two-sided smooth closed minimal hypersurface (this existence conclusion also follows from Zhou X (Ann. Math. (2), 192(3):767–820, 2020) for minmax constructions via Almgren–Pitts theory).

A central limit theorem for integrals of random waves

Abstract We derive a central limit theorem for the mean-square of random waves in the high-frequency limit over shrinking sets. Our proof applies to any compact Riemannian manifold of dimension 3 or higher, thanks to the universality of the local Weyl law. The key technical step is an estimate capturing some cancellation in a triple integral of Bessel functions, which we achieve using Gegenbauer’s addition formula.

Genericity of Continuous Maps with Positive Metric Mean Dimension

M. Gromov introduced the mean dimension for a continuous map in the late 1990’s, which is an invariant under topological conjugacy. On the other hand, the notion of metric mean dimension for a dynamical system was introduced by Lindenstrauss and Weiss in 2000 and this refines the topological entropy for dynamical systems with infinite topological entropy. In this paper we will show if N is an n dimensional compact riemannian manifold then, for any $$a\in [0,n]$$ , the set consisting of continuous maps with metric mean dimension equal to a is dense in $$C^{0}(N)$$ and for $$a=n$$ this set is residual. Furthermore, we prove some results related to the existence and, density of continuous maps, defined on Cantor sets, with positive metric mean dimension and also continous maps, defined on product spaces, with positive mean dimension.

Higher Dimensional Surgery and Steklov Eigenvalues

We show that for compact Riemannian manifolds of dimension at least 3 with nonempty boundary, we can modify the manifold by performing surgeries of codimension 2 or higher, while keeping the Steklov spectrum nearly unchanged. This shows that certain changes in the topology of a domain do not have an effect when considering shape optimization questions for Steklov eigenvalues in dimensions 3 and higher. Our result generalizes the 1-dimensional surgery in Fraser and Schoen (Adv Math 348:146–162, 2019) to higher dimensional surgeries and to higher eigenvalues. It is proved in Fraser and Schoen (Adv Math 348:146–162, 2019) that the unit ball does not maximize the first nonzero normalized Steklov eigenvalue among contractible domains in \(\mathbb {R}^n\), for \(n \ge 3\). We show that this is also true for higher Steklov eigenvalues. Using similar ideas, we show that in \(\mathbb {R}^n\), for \(n\ge 3\), the j-th normalized Steklov eigenvalue is not maximized in the limit by a sequence of contractible domains degenerating to the disjoint union of j unit balls, in contrast to the case in dimension 2 (Girouard and Polterovich in Funct Anal Appl 44:106–117, 2010).

Existence and limiting behavior of min-max solutions of the Ginzburg–Landau equations on compact manifolds

We use a natural two-parameter min-max construction to produce critical points of the Ginzburg–Landau functionals on a compact Riemannian manifold of dimension $\geq 2$. We investigate the limiting behavior of these critical points as $\varepsilon \to 0$, and show in particular that some of the energy concentrates on a nontrivial stationary, rectifiable $(n-2)$-varifold as $\varepsilon \to 0$, suggesting connections to the min-max construction of minimal $(n-2)$-submanifolds.

A Remark on the Concentration Compactness Principle in Critical Dimension

AbstractWe prove some refinements of the concentration compactness principle for Sobolev space W1, n on a smooth compact Riemannian manifold of dimension n. As an application, we extend Aubin's theorem for functions on with zero first‐order moments of the area element to the higher‐order moments case. Our arguments are very flexible and can be easily modified for functions satisfying various boundary conditions or belonging to higher‐order Sobolev spaces. © 2020 Wiley Periodicals LLC.

Evolution of the first eigenvalue of the weighted $(p,q)$-Laplacian system under rescaled Yamabe flow

Consider the triple $ \left(M, g, d\mu\right)$ as a smooth metric measure space and $ M $ is an $n$-dimensional compact Riemannian manifold without boundary, also $d\mu = e^{-f(x)}dV$ is a weighted measure. We are going to investigate the evolution problem for the first eigenvalue of the weighted $\left(p, q\right)$-Laplacian system along the rescaled Yamabe flow and we hope to find some monotonic quantities.

Nodal solutions for a Paneitz-Branson type equation

Let (M,g) be a smooth compact Riemannian manifold of dimension n≥5. Denote Lg the Paneitz-Branson type operator. In this paper, we show that there exists a nodal solution (solution with changing sign) of the nonlinear Paneitz-Branson type equation Lgv=ϵ|v|N−2v where ϵ<0. At the end, we give a geometric application of the above equation.

Second-Order Elliptic Equation with Singularities

On the compact Riemannian manifold of dimension n≥5, we study the existence and regularity of nontrivial solutions for nonlinear second-order elliptic equation with singularities. At the end, we give a geometric application of the above singular equation.

NNSC-Cobordism of Bartnik Data in High Dimensions

In this short note, we formulate three problems relating to nonnegative scalar curvature (NNSC) fill-ins. Loosely speaking, the first two problems focus on: When are $(n-1)$-dimensional Bartnik data $\big(\Sigma_i ^{n-1}, \gamma_i, H_i\big)$, $i=1,2$, NNSC-cobordant? (i.e., there is an $n$-dimensional compact Riemannian manifold $\big(\Omega^n, g\big)$ with scalar curvature $R(g)\geq 0$ and the boundary $\partial \Omega=\Sigma_{1} \cup \Sigma_{2}$ such that $\gamma_i$ is the metric on $\Sigma_i ^{n-1}$ induced by $g$, and $H_i$ is the mean curvature of $\Sigma_i$ in $\big(\Omega^n, g\big)$). If $\big(\mathbb{S}^{n-1},\gamma_{\rm std},0\big)$ is positive scalar curvature (PSC) cobordant to $\big(\Sigma_1 ^{n-1}, \gamma_1, H_1\big)$, where $\big(\mathbb{S}^{n-1}, \gamma_{\rm std}\big)$ denotes the standard round unit sphere then $\big(\Sigma_1 ^{n-1}, \gamma_1, H_1\big)$ admits an NNSC fill-in. Just as Gromov's conjecture is connected with positive mass theorem, our problems are connected with Penrose inequality, at least in the case of $n=3$. Our third problem is on $\Lambda\big(\Sigma^{n-1}, \gamma\big)$ defined below.

Pointwise Bounds for Joint Eigenfunctions of Quantum Completely Integrable Systems

Let (M, g) be a compact Riemannian manifold of dimension n and P_1:=-h^2Delta _g+V(x)-E_1 so that dp_1ne 0 on p_1=0. We assume that P_1 is quantum completely integrable (ACI) in the sense that there exist functionally independent pseuodifferential operators P_2,dots P_n with [P_i,P_j]=0, i,j=1,dots n. We study the pointwise bounds for the joint eigenfunctions, u_h of the system {P_i}_{i=1}^n with P_1u_h=E_1u_h+o(1). In Theorem 1, we first give polynomial improvements over the standard Hörmander bounds for typical points in M. In two and three dimensions, these estimates agree with the Hardy exponent h^{-frac{1-n}{4}} and in higher dimensions we obtain a gain of h^{frac{1}{2}} over the Hörmander bound. In our second main result (Theorem 3), under a real-analyticity assumption on the QCI system, we give exponential decay estimates for joint eigenfunctions at points outside the projection of invariant Lagrangian tori; that is at points xin M in the “microlocally forbidden” region p_1^{-1}(E_1)cap dots cap p_n^{-1}(E_n)cap T^*_xM=emptyset . These bounds are sharp locally near the projection of the invariant tori.

An existence theorem on the isoperimetric ratio over scalar-flat conformal classes

Let (M,g) be a smooth compact Riemannian manifold of dimension n with smooth boundary ∂M, admitting a scalar-flat conformal metric. We prove that the supremum of the isoperimetric ratio over the scalar-flat conformal class is strictly larger than the best constant of the isoperimetric inequality in the Euclidean ball, and consequently is achieved, if either (i) 9≤n≤11 and ∂M has a nonumbilic point; or (ii) 7≤n≤9, ∂M is umbilic and the Weyl tensor does not vanish identically on the boundary. This is a continuation of the work [12] by the second named author and Xiong.

The Kato smoothing effect for the nonlinear regularized Schrödinger equation on compact manifolds

<p style='text-indent:20px;'>We establish Strichartz estimates for the regularized Schrödinger equation on a two dimensional compact Riemannian manifold without boundary. As a consequence we deduce global existence and uniqueness results for the Cauchy problem for the nonlinear regularized Schrödinger equation and we prove under the geometric control condition the Kato smoothing effect for solutions of this equation in this particular geometries.

Estimates and monotonicity for a heat flow of isometric $$G_{2}$$-structures

Given a $7$-dimensional compact Riemannian manifold $\left( M,g\right) $ that admits $G_{2}$-structure, all the $G_{2}$-structures that are compatible with the metric $g$ are parametrized by unit sections of an octonion bundle over $M$. We define a natural energy functional on unit octonion sections and consider its associated heat flow. The critical points of this functional and flow precisely correspond to $G_{2}$-structures with divergence-free torsion. In this paper, we first derive estimates for derivatives of $V\left( t\right) $ along the flow and prove that the flow exists as long as the torsion remains bounded. We also prove a monotonicity formula and and an $\varepsilon $-regularity result for this flow. Finally, we show that within a metric class of $G_{2}$-structures that contains a torsion-free $G_{2}$-structure, under certain conditions, the flow will converge to a torsion-free $G_{2}$-structure.

Boundary value problems for the Brinkman system with L∞ coefficients in Lipschitz domains on compact Riemannian manifolds. A variational approach

The purpose of this paper is to show well-posedness results in L2-based Sobolev spaces for transmission, Dirichlet, Neumann, and mixed boundary value problems for the Brinkman system with L∞ coefficients in Lipschitz domains on a compact Riemannian manifold of dimension m≥2. The Dirichlet, transmission, and mixed problems for the nonlinear Darcy-Forchheimer-Brinkman system with L∞ coefficients are also analyzed. First, we focus on the well-posedness of linear transmission, Dirichlet and mixed boundary value problems for the Brinkman system with L∞ coefficients in Lipschitz domains on compact Riemannian manifolds by using a variational approach that reduces such a boundary value problem to a mixed variational formulation defined in terms of two bilinear continuous forms, one of them satisfying a coercivity condition and another one the inf-sup condition. Further, we show the equivalence between each boundary value problem for the Brinkman system with L∞ coefficients and its mixed variational counterpart, and then the well posedness in L2-based Sobolev spaces by using the Nečas-Babuška-Brezzi technique. The second goal of this paper is the construction of the Newtonian and layer potential operators for the Brinkman system with L∞ coefficients in Lipschitz domains on compact Riemannian manifolds by using the well-posedness results for the analyzed linear transmission problems. Various mapping properties of these operators are also obtained and used to describe the weak solutions of the Poisson problems with Dirichlet and Neumann conditions for the nonsmooth Brinkman system in terms of such potentials. Finally, we combine the well-posedness results of the Poisson problems of Dirichlet, transmission, and mixed type for the nonsmooth Brinkman system with a fixed point theorem in order to show the existence of a weak solution of the Poisson problem of Dirichlet, transmission, or mixed type for the (nonlinear) Darcy-Forchheimer-Brinkman system with L∞ coefficients in L2-based Sobolev spaces in Lipschitz domains on compact Riemannian manifolds of dimension m∈{2,3}.

Compactness of conformal metrics with constant Q-curvature. I

We study compactness for nonnegative solutions of the fourth order constant Q-curvature equations on smooth compact Riemannian manifolds of dimension ≥5. If the Q-curvature equals −1, we prove that all solutions are universally bounded. If the Q-curvature is 1, assuming that Paneitz operator's kernel is trivial and its Green function is positive, we establish universal energy bounds on manifolds which are either locally conformally flat (LCF) or of dimension ≤9. Moreover, assuming in addition that a positive mass type theorem holds for the Paneitz operator, we prove compactness in C4. Positive mass type theorems have been verified recently on LCF manifolds or manifolds of dimension ≤7, when the Yamabe invariant is positive. We also prove that, for dimension ≥8, the Weyl tensor has to vanish at possible blow up points of a sequence of blowing up solutions. This implies the compactness result in dimension ≥8 when the Weyl tensor does not vanish anywhere. To overcome difficulties stemming from fourth order elliptic equations, we develop a blow up analysis procedure via integral equations.