Volume growth, ``a priori'' estimates, and geometric applications
type: Item
instance of: scholarly article
title: Volume growth, ``a priori'' estimates, and geometric applications
zbMATH Open document ID: 1065.53037
author: M. Rigoli
author: S. Pigola
author: Alberto G. Setti
published in: Geometric and Functional Analysis. GAFA
publication date: 2004-05-27T00:00:00Z
review text: Let \((M,\langle\;,\;\rangle)\) be a smooth, connected, non-compact, complete Riemannian manifold of dimension \(m\geq 2\), \(\nabla\) (resp. div) the gradient (resp. divergence) operator, \(| \;| \) the norm corresponding to \(\langle\;,\;\rangle\) and the \(\varphi\)-Laplacian \(L_\varphi\) defined, for \(u\in C^1(M)\), by \(L_\varphi (u) := \text{div}(| \nabla u| ^{-1}\varphi(| \nabla u| )\nabla u)\), where \(\varphi\) is a fixed real valued function in \(C^1((0,+\infty))\cap C^0([0,+\infty))\) satisfying certain structural conditions. The Laplace-Beltrami operator [or more generally, the \(p\)-Laplacian, \(p>1\), corresponding to \(\varphi(t)=t^{p-1}\)] and the generalized mean curvature operator are natural examples of such differential operators. In the Euclidean setting the \(\varphi\)-Laplacian has been studied by many authors: \textit{J. Serrin} [Proc. Lond. Math. Soc. (3) 24, 348--366 (1972; Zbl 0229.35035); J. Funct. Anal. 5, 184--193 (1970; Zbl 0188.41701)]; \textit{L. A. Peletier} and \textit{J. Serrin} [Ann. Sc. Norm. Super. Pisa, Cl. Sci. (4) 5, 65--104 (1978; Zbl 0383.35025)]; \textit{P. Pucci, J. Serrin} and \textit{H. Zou} [J. Math Pures Appl. (9) 78, No. 8, 769--789 (1999; Zbl 0952.35045)]; \textit{P. Pucci} and \textit{J. Serrin} [J. Math. Pures Appl. (9) 79, No. 1, 57--71 (2000; Zbl 0952.35163)]; \textit{B. Franchi, E. Lanconelli} and \textit{J. Serrin} [Adv. Math. 118, No. 2, 177--243 (1996; Zbl 0853.35035)]. The authors establish ``a priori'' estimates for solutions of differential equations or inequalities of the form \(L_\varphi(u)\geq F(x,u)\), \(x\in M\), under volume growth assumptions of geodesic balls only. In particular, the weak maximum principle for \(L_\varphi\), previously obtained by the last two authors is derived [Rev. Mat. Iberoam. 17, No. 3, 471--520 (2001; Zbl 1043.58022)]. As consequences, Schwarz type lemmas for holomorphic maps between complex manifolds of the same dimension and for conformal maps, a nonexistence result fo…
Mathematics Subject Classification ID: 53C21
Mathematics Subject Classification ID: 58J05
Mathematics Subject Classification ID: 58J60
Mathematics Subject Classification ID: 35B50
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 35R70
zbMATH DE Number: 2067277
zbMATH Keywords: complete Riemannian manifold
zbMATH Keywords: \(\varphi\)-Laplacian
zbMATH Keywords: volume growth
zbMATH Keywords: geodesic ball
zbMATH Keywords: a priori estimate
zbMATH Keywords: maximum principle
zbMATH Keywords: Schwarz lemma
zbMATH Keywords: Yamabe equation
zbMATH Keywords: Liouville theorem
reviewed by: Mircea Craioveanu
MaRDI profile type: Publication
full work available at URL: s00039-003-0447-2
OpenAlex ID: W2025853708
DOI: 10.1007/S00039-003-0447-2
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description: scientific article; zbMATH DE number 2067277