The concept and criteria of the capacitive type of the non-compact Riemannian manifold based on the generalized capacity

V. M. Keselman

Matematičeskaâ fizika i kompʹûternoe modelirovanie, Tome 22 (2019) no. 2, pp. 21-32.

Résumé

Let $M^n$ be a non-compact $n$-dimensional Riemannian manifold and let $p>1$ be a fixed real number. We call $(F, p)$-capacity of a compact set $K\subset M^n$ a value $\inf\int_{M^n}(F(x, \nabla u))^p dv$, where the exact lower bound is taken over all smooth functions $u$ finite in $M^n$ and such that $u \geq 1$ on $K$. Function $F = F(x, \xi)$, $(x, \xi)\in TM^n$ is smooth, non-negative and satisfies certain general conditions. A special case of $(F, p)$-capacity is, e. g., the conformal capacity when $F(x, \xi) = |\xi|$ and $p = n$. We based this notion of $(F, p)$-capacity on the work of G. Choquet, V.G. Mazya, and V.M. Miklyukov. Let us introduce the concept of the type of a non-compact manifold $M^n$ as follows. We say that $M^n$ is of $(F, p)$-parabolic type, if the $(F, p)$-capacity of some non-degenerate compact $K\subset M^n$ is zero. Otherwise, we say that manifold $M^n$ is of $(F, p)$-hyperbolic type. Like in the classical case, this notion of $(F, p)$-type of the non-compact Riemannian manifold is invariant with respect to the specific choice of the compact set $K$. We prove the criteria for the manifold to be of $(F, p)$-parabolic or $(F, p)$-hyperbolic type. Special cases of these are the well-known criteria of conformal type of a Riemannian manifold expressed in terms of growth of the volume $V(r)$ of geodesic balls or of area $S(r)$ of their boundary spheres of radius $r$. In the general case of criteria of $(F, p)$-type of manifold $M^n$ the role of the class of complete metrics conformal to the initial metric of the manifold takes on the class of exhaustion functions $h$ of manifold $M^n$, and the roles of $V(r)$ and $S(r)$ are taken by functions $V_{F, p, h}(r)=\int_{h\leq r}(F(x,\nabla h))^p dv$ and $ S_{F, p, h}(r) = \int_{h = r}(F(x, \nabla h))^p(d\sigma /|\nabla h|)$, respectively. The criteria themselves are expressed in terms of the growth of these functions. For instance, the following conditions $$\int^{+\infty}\left(\frac{r}{V_{F, p, h}(r)}\right)^{\frac{1}{p-1}}dr = \infty, \; \int^{+\infty}\left(\frac{1}{S_{F, p, h}(r)}\right)^{\frac{1}{p-1}}dr = \infty $$ characterize the $(F, p)$-parabolic type of the non-compact Riemannian manifold.

Détail

Keywords: Riemannian manifold, capacity, p-hyperbolic type, volume of a geodesic ball, area of the geodesic sphere, exhaustion function.
Mots-clés : conformal type, p-parabolictype