caa a22 4500 469117338 CHVBK 20180323133128.0 cr unu---uuuuu 170328e19920501xx s 000 0 eng 10.1007/BF02921293 doi (NATIONALLICENCE)springer-10.1007/BF02921293 On the solutions to certain laplace inequalities with applications to geometry of submanifolds [Elektronische Daten] [Leslie Coghlan, Yoe Itokawa, Roman Kosecki] LetM be a complete Riemannian manifold with Ricci curvature bounded from below. We give an explicit estimate for the size of the negative sets of solutions to the differential inequality Δu ≥λu where Δ is the Laplacian and λ is a negative constant. This inequality arises naturally when we study the lengthH of the mean curvature of an isometric immersionf of M into another Riemannian manifoldN with curvature bounded above by some constantκ. Suppose that the image f(M) does not meet the cut locus of some pointo ∈ N. As a consequence of our estimate, we prove that, givenρ > 0, if supH is less than a certain explicit expression μ(κ, ρ) inρ andκ on any domainU that contains an inscribed ball of radius greater than an explicitly computable numberR, then the diameter of the setf(U) inN must exceed 2ρ. Moreover, if supH = μ(κ, ρ) onM and the diameter off(M) inN equals 2ρ, thenf is a minimal immersion into a distance sphere of radiusρ inN. CRC Press, Inc, 1992 Eigenvalue problem nationallicence isometric immersion nationallicence Laplace-Beltrami operator nationallicence mean curvature vector nationallicence Riemannian manifold nationallicence scalar curvature nationallicence sectional curvature nationallicence subsolution nationallicence Coghlan Leslie Department of Mathematics, University of Alabama at Birmingham, 35205, Birmingham, AL, USA aut Itokawa Yoe Department of Mathematics, University of Alabama at Birmingham, 35205, Birmingham, AL, USA aut Kosecki Roman Department of Mathematics, University of Alabama at Birmingham, 35205, Birmingham, AL, USA aut The Journal of Geometric Analysis Springer-Verlag 2/3(1992-05-01), 195-211 1050-6926 2:3<195 1992 2 12220 https://doi.org/10.1007/BF02921293 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/BF02921293 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Coghlan Leslie Department of Mathematics, University of Alabama at Birmingham, 35205, Birmingham, AL, USA aut NATIONALLICENCE 700 1- Itokawa Yoe Department of Mathematics, University of Alabama at Birmingham, 35205, Birmingham, AL, USA aut NATIONALLICENCE 700 1- Kosecki Roman Department of Mathematics, University of Alabama at Birmingham, 35205, Birmingham, AL, USA aut NATIONALLICENCE 773 0- The Journal of Geometric Analysis Springer-Verlag 2/3(1992-05-01), 195-211 1050-6926 2:3<195 1992 2 12220 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer