Journal of Geometric Analysis最新文献

We investigate the geometry of the space of immersed closed curves equipped with reparametrization-invariant Riemannian metrics; the metrics we consider are Sobolev metrics of possible fractional-order $q\in [0,\infty )$. We establish the critical Sobolev index on the metric for several key geometric properties. Our first main result shows that the Riemannian metric induces a metric space structure if and only if $q>1/2$. Our second main result shows that the metric is geodesically complete (i.e., the geodesic equation is globally well posed) if $q>3/2$, whereas if $q<3/2$ then finite-time blowup may occur. The geodesic completeness for $q>3/2$ is obtained by proving metric completeness of the space of $H^q$-immersed curves with the distance induced by the Riemannian metric.

Suppose that $\Omega \subset R^{n+1}$, $n\ge 1$, is a uniform domain with n-Ahlfors regular boundary and L is a (not necessarily symmetric) divergence form elliptic, real, bounded operator in $\Omega$. We show that the corresponding elliptic measure ${\omega }_L$ is quantitatively absolutely continuous with respect to surface measure of $\partial \Omega$ in the sense that ${\omega }_L\in A_{\infty }\left(\sigma \right)$ if and only if any bounded solution u to $Lu=0$ in $\Omega$ is $\epsilon$-approximable for any $\epsilon \in (0,1)$. By $\epsilon$-approximability of u we mean that there exists a function $\Phi ={\Phi }^{\epsilon }$ such that ${\Vert u-\Phi \Vert }_{L^{\infty }\left(\Omega \right)}\le \epsilon {\Vert u\Vert }_{L^{\infty }\left(\Omega \right)}$ and the measure ${\tilde{\mu }}_{\Phi }$ with $d\tilde{\mu }=\left|\nabla \Phi \left(Y\right)\right|dY$ is a Carleson measure with $L^{\infty }$ control over the Carleson norm. As a consequence of this approximability result, we show that boundary $\text{BMO}$ functions with compact support can have Varopoulos-type extensions even in some sets with unrectifiable boundaries, that is, smooth extensions that converge non-tangentially back to the original data and that satisfy $L^1$-type Carleson measure estimates with $\text{BMO}$ control over the Carleson norm. Our result complements the recent work of Hofmann and the third named author who showed the existence of these types of extensions in the presence of a quantitative rectifiability hypothesis.

We show that Worm domains are not Gromov hyperbolic with respect to the Kobayashi distance.

This article is concerned with the stability of triharmonic maps and in particular triharmonic hypersurfaces. After deriving a number of general statements on the stability of triharmonic maps we focus on the stability of triharmonic hypersurfaces in space forms, where we pay special attention to their normal stability. We show that triharmonic hypersurfaces of constant mean curvature in Euclidean space are weakly stable with respect to normal variations while triharmonic hypersurfaces of constant mean curvature in hyperbolic space are stable with respect to normal variations. For the case of a spherical target we show that the normal index of the small proper triharmonic hypersphere $\varphi :S^m(1/\sqrt{3})\hookrightarrow S^{m+1}$ is equal to one and make some comments on the normal stability of the proper triharmonic Clifford torus.

The aim of this paper is to give a thorough insight into the relationship between the Rumin complex on Carnot groups and the spectral sequence obtained from the filtration on forms by homogeneous weights that computes the de Rham cohomology of the underlying group.

In this paper, we introduce the notion of horizontally affine, h-affine in short, function and give a complete description of such functions on step-2 Carnot algebras. We show that the vector space of h-affine functions on the free step-2 rank-n Carnot algebra is isomorphic to the exterior algebra of $R^n$. Using that every Carnot algebra can be written as a quotient of a free Carnot algebra, we shall deduce from the free case a description of h-affine functions on arbitrary step-2 Carnot algebras, together with several characterizations of those step-2 Carnot algebras where h-affine functions are affine in the usual sense of vector spaces. Our interest for h-affine functions stems from their relationship with a class of sets called precisely monotone, recently introduced in the literature, as well as from their relationship with minimal hypersurfaces.

In this paper, we introduce the topological state derivative for general topological dilatations and explore its relation to standard optimal control theory. We show that for a class of partial differential equations, the shape-dependent state variable can be differentiated with respect to the topology, thus leading to a linearised system resembling those occurring in standard optimal control problems. However, a lot of care has to be taken when handling the regularity of the solutions of this linearised system. In fact, we should expect different notions of (very) weak solutions, depending on whether the main part of the operator or its lower order terms are being perturbed. We also study the relationship with the topological state derivative, usually obtained through classical topological expansions involving boundary layer correctors. A feature of the topological state derivative is that it can either be derived via Stampacchia-type regularity estimates or alternately with classical asymptotic expansions. It should be noted that our approach is flexible enough to cover more than the usual case of point perturbations of the domain. In particular, and in the line of (Delfour in SIAM J Control Optim 60(1):22-47, 2022; J Convex Anal 25(3):957-982, 2018), we deal with more general dilatations of shapes, thereby yielding topological derivatives with respect to curves, surfaces or hypersurfaces. To draw the connection to usual topological derivatives, which are typically expressed with an adjoint equation, we show how usual first-order topological derivatives of shape functionals can be easily computed using the topological state derivative.

We prove a Bochner-Kodaira-Nakano formula and establish Szegő kernel expansions on complete strictly pseudoconvex CR manifolds with transversal CR $R$ -action under certain natural geometric conditions. As a consequence we show that such manifolds are locally CR embeddable.

We propose an analytic torsion for the Rumin complex associated with generic rank two distributions on closed 5-manifolds. This torsion behaves as expected with respect to Poincaré duality and finite coverings. We establish anomaly formulas, expressing the dependence on the sub-Riemannian metric and the 2-plane bundle in terms of integrals over local quantities. For certain nilmanifolds, we are able to show that this torsion coincides with the Ray-Singer analytic torsion, up to a constant.

Let $X\to P^1$ be an elliptically fibered K3 surface, admitting a sequence ${\omega }_i$ of Ricci-flat metrics collapsing the fibers. Let V be a holomorphic SU(n) bundle over X, stable with respect to ${\omega }_i$ . Given the corresponding sequence ${\Xi }_i$ of Hermitian-Yang-Mills connections on V, we prove that, if E is a generic fiber, the restricted sequence ${\Xi }_i{|}_E$ converges to a flat connection $A_0$ . Furthermore, if the restriction ${V|}_E$ is of the form ${\oplus }_{j=1}^n{O}_E(q_j-0)$ for n distinct points $q_j\in E$ , then these points uniquely determine $A_0$ .