Volume growth, ``a priori'' estimates, and geometric applications
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 for positive solutions of the non-compact Yamabe equation under volume growth conditions, a vanishing result for bounded harmonic forms in the middle dimension on locally conformally flat complete manifolds of even dimension with positive scalar curvature, a Liouville-type theorem, as well as other applications, are deduced.
scientific article; zbMATH DE number 2067277
scientific article