Journal of Geometric Analysis最新文献

A family of sharp $L^p$ Sobolev inequalities is established by averaging the length of i-dimensional projections of the gradient of a function. Moreover, it is shown that each of these new inequalities directly implies the classical $L^p$  Sobolev inequality of Aubin and Talenti and that the strongest member of this family is the only affine invariant one among them-the affine $L^p$  Sobolev inequality of Lutwak, Yang, and Zhang. When $p=1$ , the entire family of new Sobolev inequalities is extended to functions of bounded variation to also allow for a complete classification of all extremal functions in this case.

The goal of the paper is to sharpen and generalise bounds involving Cheeger's isoperimetric constant h and the first eigenvalue ${\lambda }_1$ of the Laplacian. A celebrated lower bound of ${\lambda }_1$ in terms of h, ${\lambda }_1\ge h^2/4$ , was proved by Cheeger in 1970 for smooth Riemannian manifolds. An upper bound on ${\lambda }_1$ in terms of h was established by Buser in 1982 (with dimensional constants) and improved (to a dimension-free estimate) by Ledoux in 2004 for smooth Riemannian manifolds with Ricci curvature bounded below. The goal of the paper is twofold. First: we sharpen the inequalities obtained by Buser and Ledoux obtaining a dimension-free sharp Buser inequality for spaces with (Bakry-Émery weighted) Ricci curvature bounded below by $K\in R$ (the inequality is sharp for $K>0$ as equality is obtained on the Gaussian space). Second: all of our results hold in the higher generality of (possibly non-smooth) metric measure spaces with Ricci curvature bounded below in synthetic sense, the so-called $\mathrm{RCD}(K,\infty )$ spaces.

The reflection map introduced by D'Angelo is applied to deduce simpler descriptions of nondegeneracy conditions for sphere maps and to the study of infinitesimal deformations of sphere maps. It is shown that the dimension of the space of infinitesimal deformations of a nondegenerate sphere map is bounded from above by the explicitly computed dimension of the space of infinitesimal deformations of the homogeneous sphere map. Moreover a characterization of the homogeneous sphere map in terms of infinitesimal deformations is provided.

A contact twisted cubic structure $(M,C,\gamma )$ is a 5-dimensional manifold $M$ together with a contact distribution $C$ and a bundle of twisted cubics $\gamma \subset P\left(C\right)$ compatible with the conformal symplectic form on $C$ . The simplest contact twisted cubic structure is referred to as the contact Engel structure; its symmetry group is the exceptional group $G_2$ . In the present paper we equip the contact Engel structure with a smooth section $\sigma :M\to \gamma$ , which "marks" a point in each fibre ${\gamma }_x$ . We study the local geometry of the resulting structures $(M,C,\gamma ,\sigma )$ , which we call marked contact Engel structures. Equivalently, our study can be viewed as a study of foliations of $M$ by curves whose tangent directions are everywhere contained in $\gamma$ . We provide a complete set of local invariants of marked contact Engel structures, we classify all homogeneous models with symmetry groups of dimension $\ge 6$ up to local equivalence, and we prove an analogue of the classical Kerr theorem from Relativity.

We minimize a linear combination of the length and the $L^2$ -norm of the curvature among networks in $R^d$ belonging to a given class determined by the number of curves, the order of the junctions, and the angles between curves at the junctions. Since this class lacks compactness, we characterize the set of limits of sequences of networks bounded in energy, providing an explicit representation of the relaxed problem. This is expressed in terms of the new notion of degenerate elastic networks that, rather surprisingly, involves only the properties of the given class, without reference to the curvature. In the case of $d=2$ we also give an equivalent description of degenerate elastic networks by means of a combinatorial definition easy to validate by a finite algorithm. Moreover we provide examples, counterexamples, and additional results that motivate our study and show the sharpness of our characterization.