Euclidean Setting Research Articles

Rigidities of isoperimetric inequality under nonnegative Ricci curvature

The sharp isoperimetric inequality for non-compact Riemannian manifolds with nonnegative Ricci curvature and Euclidean volume growth has been obtained in increasing generality with different approaches in a number of contributions culminated by Balogh and Kristály (2023) also covering metric-measure spaces satisfying the nonnegative Ricci curvature condition in the synthetic sense of Lott, Sturm and Villani. In sharp contrast with the compact case of positive Ricci curvature, for a large class of spaces including weighted Riemannian manifolds, no complete characterization of the equality cases is present in the literature. The scope of this paper is to settle this problem by proving, in the same generality as Balogh and Kristály (2023), that the equality in the isoperimetric inequality can be attained only by metric balls. Whenever this happens the space is forced, in a measure theoretic sense, to be a cone. Our result applies to different frameworks yielding as corollaries new rigidity results: it extends the rigidity results of Brendle (2023) for weighted Riemannian manifolds and the rigidity results of Antonelli et al. (2023) for general \mathsf{RCD} spaces. It also applies to the Euclidean setting by proving that optimizers in the anisotropic and weighted isoperimetric inequality for Euclidean cones are necessarily the Wulff shapes.

Circulating Gut Microbe-Derived Metabolites Are Associated with Hepatocellular Carcinoma.

Forty-one patients with HCC and 96 healthy controls were enrolled from surgical clinics at the Cleveland Clinic from 2016 to 2020. Gut microbiota-derived circulating metabolites detectable in plasma were compared between patients with HCC and healthy controls. Hierarchical clustering was performed for generating heatmaps based on circulating metabolite concentrations using ClustVis, with Euclidean and Ward settings and significant differences between metabolite concentrations were tested using a binary logistic regression model. In patients with HCC, 25 (61%) had histologically confirmed cirrhosis. Trimethylamine (TMA)-related metabolites were found at higher concentrations in those with HCC, including choline (p < 0.001), betaine (p < 0.001), carnitine (p = 0.007), TMA (p < 0.001) and trimethylamine N-oxide (TMAO, p < 0.001). Notably, concentrations of P-cresol glucuronide (p < 0.001), indole-lactic acid (p = 0.038), 5-hydroxyindoleacetic acid (p < 0.0001) and 4-hydroxyphenyllactic acid (p < 0.001) were also increased in those with HCC compared to healthy controls. Hierarchical clustering of the metabolite panel separated patients based on the presence of HCC (p < 0.001), but was not able to distinguish between patients with HCC based on the presence of cirrhosis (p = 0.42). Gut microbiota-derived metabolites were differentially abundant in patients with HCC versus healthy controls. The observed perturbations of the TMAO pathway in HCC seem particularly promising as a target of future research and may have both diagnostic and therapeutic implications.

Splines on manifolds: A survey

Splines in the manifold setting have been defined as extensions from the standard Euclidean setting, but they are far more complicated. Alternative approaches, which are equivalent in the Euclidean case, lead to different results in the manifold case; the existence conditions are often quite restrictive; and the necessary computations are rather involved. All difficulties stem from the peculiar nature of the geodesic distance: in general, shortest geodesics may be not unique and the dependence on their endpoints may not be smooth; and distances cannot be computed in closed form. The former issue may impose strong limitations on the placement of control points. While the latter may greatly complicate the computations. Nevertheless, some recent results suggest that splines on surfaces may have practical impact on CAGD applications. We review the literature on this topic, accounting for both theoretical results and practical implementations.

The Brownian transport map

Contraction properties of transport maps between probability measures play an important role in the theory of functional inequalities. The actual construction of such maps, however, is a non-trivial task and, so far, relies mostly on the theory of optimal transport. In this work, we take advantage of the infinite-dimensional nature of the Gaussian measure and construct a new transport map, based on the Föllmer process, which pushes forward the Wiener measure onto probability measures on Euclidean spaces. Utilizing the tools of the Malliavin and stochastic calculus in Wiener space, we show that this Brownian transport map is a contraction in various settings where the analogous questions for optimal transport maps are open. The contraction properties of the Brownian transport map enable us to prove functional inequalities in Euclidean spaces, which are either completely new or improve on current results. Further and related applications of our contraction results are the existence of Stein kernels with desirable properties (which lead to new central limit theorems), as well as new insights into the Kannan–Lovász–Simonovits conjecture. We go beyond the Euclidean setting and address the problem of contractions on the Wiener space itself. We show that optimal transport maps and causal optimal transport maps (which are related to Brownian transport maps) between the Wiener measure and other target measures on Wiener space exhibit very different behaviors.

Obstacle Problems with Double Boundary Condition for Least Gradient Functions in Metric Measure Spaces

AbstractIn the setting of a metric space equipped with a doubling measure supporting a (1, 1)-Poincaré inequality, we study the problem of minimizing the BV-energy in a bounded domain $$\Omega $$ Ω of functions bounded between two obstacle functions inside $$\Omega $$ Ω , and whose trace lies between two prescribed functions on the boundary. If the class of candidate functions is nonempty, we show that solutions exist for continuous obstacles and continuous boundary data when $$\Omega $$ Ω is a uniform domain whose boundary is of positive mean curvature in the sense of Lahti, Malý, Shanmugalingam, and Speight (2019). While such solutions are not unique in general, we show the existence of unique minimal solutions. Since candidate functions need not agree outside of the domain, standard compactness arguments fail to provide existence of weak solutions as they are defined for the problem with single boundary condition. To overcome this, we introduce a class of $$ \varepsilon $$ ε -weak solutions as an intermediate step. Our existence results generalize those of Ziemer and Zumbrun (1999), who studied this problem in the Euclidean setting with a single obstacle and single boundary condition.

Asymptotic behavior of solutions to the extension problem for the fractional Laplacian on noncompact symmetric spaces

This work deals with the extension problem for the fractional Laplacian on Riemannian symmetric spaces G/K of noncompact type and of general rank, which gives rise to a family of convolution operators, including the Poisson operator. More precisely, motivated by Euclidean results for the Poisson semigroup, we study the long-time asymptotic behavior of solutions to the extension problem for L1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^1$$\\end{document} initial data. In the case of the Laplace–Beltrami operator, we show that if the initial data are bi-K-invariant, then the solution to the extension problem behaves asymptotically as the mass times the fundamental solution, but this convergence may break down in the non-bi-K-invariant case. In the second part, we investigate the long-time asymptotic behavior of the extension problem associated with the so-called distinguished Laplacian on G/K. In this case, we observe phenomena which are similar to the Euclidean setting for the Poisson semigroup, such as L1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^1$$\\end{document} asymptotic convergence without the assumption of bi-K-invariance.

Isoperimetric inequalities and regularity of A-harmonic functions on surfaces

We investigate the logarithmic and power-type convexity of the length of the level curves for a-harmonic functions on smooth surfaces and related isoperimetric inequalities. In particular, our analysis covers the p-harmonic and the minimal surface equations. As an auxiliary result, we obtain higher Sobolev regularity properties of the solutions, including the W2,2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$W^{2,2}$$\\end{document} regularity. The results are complemented by a number of estimates for the derivatives L′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L'$$\\end{document} and L′′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L''$$\\end{document} of the length of the level curve function L, as well as by examples illustrating the presentation. Our work generalizes results due to Alessandrini, Longinetti, Talenti and Lewis in the Euclidean setting, as well as a recent article of ours devoted to the harmonic case on surfaces.

Asymptotic covariances for functionals of weakly stationary random fields

Let (Ax)x∈Rd be a measurable, weakly stationary random field, i.e. E[Ax]=E[Ay], Cov(Ax,Ay)=K(x−y), ∀x,y∈Rd, with covariance function K:Rd→R.Assuming only that the integral covariance functionwt≔∫{|z|≤t}K(z)dz is regularly varying (which encompasses the classical assumptions found in the literature), we compute limt→∞Cov∫tDAxdxtd/2wt1/2,∫tLAydytd/2wt1/2 for D,L⊆Rd belonging to a certain class of compact sets.As an application, we combine this result with existing limit theorems to obtain multi-dimensional limit theorems for non-linear functionals of stationary Gaussian fields, in particular proving new results for the Berry’s random wave model. At the end of the paper, we also show how the problem for A with a general continuous covariance function K can be reduced to the same problem for a radial, continuous covariance function Kiso.The novel ideas of this work are mainly based on regularity conditions for (cross) covariograms of Euclidean sets and standard properties of regularly varying functions.

Universal consistency of the k-NN rule in metric spaces and Nagata dimension. II

We continue to investigate the k nearest neighbour (k-NN) learning rule in complete separable metric spaces. Thanks to the results of Cérou and Guyader (2006) and Preiss (1983), this rule is known to be universally consistent in every such metric space that is sigma-finite dimensional in the sense of Nagata. Here we show that the rule is strongly universally consistent in such spaces in the absence of ties. Under the tie-breaking strategy applied by Devroye, Györfi, Krzyżak, and Lugosi (1994) in the Euclidean setting, we manage to show the strong universal consistency in non-Archimedian metric spaces (i.e., those of Nagata dimension zero). Combining the theorem of Cérou and Guyader with results of Assouad and Quentin de Gromard (2006), one deduces that the k-NN rule is universally consistent in metric spaces having finite dimension in the sense of de Groot. In particular, the k-NN rule is universally consistent in the Heisenberg group which is not sigma-finite dimensional in the sense of Nagata as follows from an example independently constructed by Korányi and Reimann (1995) and Sawyer and Wheeden (1992).

Complex powers of the wave operator and the spectral action on Lorentzian scattering spaces

We consider perturbations of Minkowski space as well as more general spacetimes on which the wave operator \square_{g} is known to be essentially self-adjoint. We define complex powers (\square_{g}-i\varepsilon)^{-\alpha} by functional calculus, and show that the trace density exists as a meromorphic function of \alpha . We relate its poles to geometric quantities, in particular to the scalar curvature. The results allow us to formulate a spectral action principle which serves as a simple Lorentzian model for the bosonic part of the Chamseddine–Connes action. Our proof combines microlocal resolvent estimates, including radial propagation estimates, with uniform estimates for the Hadamard parametrix. The arguments work in Lorentzian signature directly and do not rely on transition from the Euclidean setting.

Blow-up for semilinear parabolic equations in cones of the hyperbolic space

We investigate existence and nonexistence of global in time nonnegative solutions to the semilinear heat equation, with a reaction term of the type posed on cones of the hyperbolic space. Under a certain assumption on µ and p, related to the bottom of the spectrum of in , we prove that any solution blows up in finite time, for any nontrivial nonnegative initial datum. Instead, if the parameters µ and p satisfy the opposite condition we have: (a) blow-up when the initial datum is large enough, (b) existence of global solutions when the initial datum is small enough. Hence our conditions on the parameters µ and p are optimal. We see that blow-up and global existence do not depend on the amplitude of the cone. This is very different from what happens in the Euclidean setting (Bandle and Levine 1989 Trans. Am. Math. Soc. 316 595–622), and it is essentially due to a specific geometric feature of .

Mixed weak‐type inequalities in Euclidean spaces and in spaces of the homogeneous type

AbstractIn this paper, we provide mixed weak‐type inequalities generalizing previous results in an earlier work by Caldarelli and the second author and also in the spirit of earlier results by Lorente et al. One of the main novelties is that, besides obtaining estimates in the Euclidean setting, results are provided as well in spaces of the homogeneous type, being the first mixed weak‐type estimates that we are aware of in that setting.

Topological regularity of isoperimetric sets in PI spaces having a deformation property

We prove topological regularity results for isoperimetric sets in PI spaces having a suitable deformation property, which prescribes a control on the increment of the perimeter of sets under perturbations with balls. More precisely, we prove that isoperimetric sets are open, satisfy boundary density estimates and, under a uniform lower bound on the volumes of unit balls, are bounded. Our results apply, in particular, to the class of possibly collapsed $\mathrm {RCD}(K,N)$ spaces. As a consequence, the rigidity in the isoperimetric inequality on possibly collapsed $\mathrm {RCD}(0,N)$ spaces with Euclidean volume growth holds without the additional assumption on the boundedness of isoperimetric sets. Our strategy is of interest even in the Euclidean setting, as it simplifies some classical arguments.

Discrete heat equation with irregular thermal conductivity and tempered distributional data

In this paper, we consider a semi-classical version of the nonhomogeneous heat equation with singular time-dependent coefficients on the lattice $\hbar \mathbb {Z}^n$ . We establish the well-posedness of such Cauchy problems in the classical sense when regular coefficients are considered, and analyse how the notion of very weak solution adapts in such equations when distributional coefficients are regarded. We prove the well-posedness of both the classical and the very weak solution in the weighted spaces $\ell ^{2}_{s}(\hbar \mathbb {Z}^n)$ , $s \in \mathbb {R}$ , which is enough to prove the well-posedness in the space of tempered distributions $\mathcal {S}'(\hbar \mathbb {Z}^n)$ . Notably, when $s=0$ , we show that for $\hbar \rightarrow 0$ , the classical (resp. very weak) solution of the heat equation in the Euclidean setting $\mathbb {R}^n$ is recaptured by the classical (resp. very weak) solution of it in the semi-classical setting $\hbar \mathbb {Z}^n$ .

Nonlocal estimates for the volume preserving mean curvature flow and applications

We obtain estimates on nonlocal quantities appearing in the volume preserving mean curvature flow (VPMCF) in the closed, Euclidean setting. As a result we demonstrate that blowups of finite time type I singularities of VPMCF are ancient solutions to mean curvature flow (MCF), prove that monotonicity methods may always be applied at these finite times and obtain information on the asymptotics of the flow. In the case of type II singularities, asymptotic flows corresponding to ’Hamilton’s rescaling procedures’ are eternal solutions of the MCF.

Ruled surfaces as translating solitons of the inverse mean curvature flow in the three-dimensional Lorentz-Minkowski space

In this paper, we classify the nondegenerate ruled surfaces in the three-dimensional Lorentz-Minkowski space that are translating solitons for the inverse mean curvature flow. In particular, we prove the existence of non-cylindrical ruled translating solitons, which contrast with the Euclidean setting.

Decentralized Riemannian Algorithm for Nonconvex Minimax Problems

The minimax optimization over Riemannian manifolds (possibly nonconvex constraints) has been actively applied to solve many problems, such as robust dimensionality reduction and deep neural networks with orthogonal weights (Stiefel manifold). Although many optimization algorithms for minimax problems have been developed in the Euclidean setting, it is difficult to convert them into Riemannian cases, and algorithms for nonconvex minimax problems with nonconvex constraints are even rare. On the other hand, to address the big data challenges, decentralized (serverless) training techniques have recently been emerging since they can reduce communications overhead and avoid the bottleneck problem on the server node. Nonetheless, the algorithm for decentralized Riemannian minimax problems has not been studied. In this paper, we study the distributed nonconvex-strongly-concave minimax optimization problem over the Stiefel manifold and propose both deterministic and stochastic minimax methods. The Steifel manifold is a non-convex set. The global function is represented as the finite sum of local functions. For the deterministic setting, we propose DRGDA and prove that our deterministic method achieves a gradient complexity of O( epsilon(-2)) under mild conditions. For the stochastic setting, we propose DRSGDA and prove that our stochastic method achieves a gradient complexity of O( epsilon(-4)). The DRGDA and DRSGDA are the first algorithms for distributed minimax optimization with nonconvex constraints with exact convergence. Extensive experimental results on the Deep Neural Networks (DNNs) training over the Stiefel manifold demonstrate the efficiency of our algorithms.

Embedding bipartite distance graphs under Hamming metric in finite fields

The famous Erdős-Falconer distance problems aim to quantify the extent to which large sets must determine many distinct distances. The problems in both the Euclidean setting and the discrete setting have received much attention. An interesting generalization of Erdős-Falconer problems is to find locating point configurations with prescribed metric structure within large subsets. In this paper, we consider such problem in the discrete setting. We say a distance graph H in Fqn under Hamming metric means a graph H=(V,E) with vertices as the points in V⊆Fqn and edges being assigned by Hamming distance between the pairs of vertices. We call a subset A⊆Fqn contains an isometric copy of distance graph H, if there exists an embedding ϕ:V(H)→A such that for every edge uv∈E(H), the Hamming distance between ϕ(u) and ϕ(v) is equal to the Hamming distance between u and v. When q is a fixed prime power and n goes to infinity, we show that for any given bipartite graph H with ex(n,H)=O(n), any subset A⊆Fqn contains a positive proportion of all possible isometric copies of distance graphs H whose edges are assigned by the same Hamming distance, if |A|>q(1−c)n for some c=c(q,H)>0. Our methods also work for general bipartite graphs. Moreover, we will give the quantitative bounds for the above constant c=c(q,H).Unlike using analytical method as before, our ideas are more combinatorial, which mainly include two techniques. The first one is the celebrated dependent random choice, which is like glue, sticking several small structures together. The second one is a new extension of the modular version of the classical Delsarte's inequality. In particular, this extension itself is of independent interest, which also helps us build a surprising connection between our main results and the extremal number of the bipartite graphs.

Cohomology of annuli, duality and L∞-differential forms on Heisenberg groups

In the last few years the authors proved Poincaré and Sobolev type inequalities in Heisenberg groups Hn for differential forms in the Rumin's complex. The need to substitute the usual de Rham complex of differential forms for Euclidean spaces with the Rumin's complex is due to the different stratification of the Lie algebra of Heisenberg groups. The crucial feature of Rumin's complex is that dc is a differential operator of order 1 or 2 according to the degree of the form.Roughly speaking, Poincaré and Sobolev type inequalities are quantitative formulations of the well known topological problem whether a closed form is exact. More precisely, for suitable p and q, we mean that every exact differential form ω in Lp admits a primitive ϕ in Lq such that ‖ϕ‖Lq≤C‖ω‖Lp. The cases of the norm Lp, p≥1 and q<∞ have been already studied in a series of papers by the authors. In the present paper we deal with the limiting case where q=∞: it is remarkable that, unlike in the scalar case, when the degree of the forms ω is at least 2, we can take q=∞ in the left-hand side of the inequality. The corresponding inequality in the Euclidean setting RN (p=N and q=∞) was proven by Bourgain & Brezis.

Schrödinger equation on non-compact symmetric spaces

We establish sharp-in-time kernel and dispersive estimates for the Schrödinger equation on non-compact Riemannian symmetric spaces of any rank. Due to the particular geometry at infinity and the Kunze-Stein phenomenon, these properties are more pronounced in large time and enable us to prove the global-in-time Strichartz inequality for a larger family of admissible couples than in the Euclidean case. Consequently, we obtain the global well-posedness for the corresponding semilinear equation with lower regularity data and some scattering properties for small powers which are known to fail in the Euclidean setting. The crucial kernel estimates are achieved by combining the stationary phase method based on a subtle barycentric decomposition, a subordination formula of the Schrödinger group to the wave propagator and an improved Hadamard parametrix.