A Note on the Critical Laplace Equation and Ricci Curvature

Abstract

We study strictly positive solutions to the critical Laplace equation -Δu=n(n-2)un+2n-2,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} - \\Delta u = n(n-2) u^{\\frac{n+2}{n-2}}, \\end{aligned}$$\\end{document}decaying at most like d(o,x)-(n-2)/2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d(o, x)^{-(n-2)/2}$$\\end{document}, on complete noncompact manifolds (M, g) with nonnegative Ricci curvature, of dimension n≥3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n \\ge 3$$\\end{document}. We prove that, under an additional mild assumption on the volume growth, such a solution does not exist, unless (M, g) is isometric to Rn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb R^n$$\\end{document} and u is a Talenti function. The method employs an elementary analysis of a suitable function defined along the level sets of u.