Connecting Poincaré Inequality with Sobolev Inequalities on Riemannian Manifolds

We connect the Poincar & eacute; inequality with the Sobolev inequality on Riemannian manifold in a family of integral inequalities (1.5). For these continuum of inequalities, we obtain topological structure theorems of manifolds generalizing previous unification theorems in both intrinsic and extrinsic settings ([33]). Manifolds which admit any of these integral inequalities are nonparabolic, affect topology, geometry, analysis, and admit nonconstant bounded harmonic functions of finite energy. As a consequence, we have proven a Conjecture of Schoen-Yau ([27, p.74]) to be true in dimension two with hypotheses weaker than that used in [1] and [33]( which were weaker than the hypotheses set in the conjecture, ( cf. Remark 1.5). In the same philosophy and spirit as in ([31]), we prove that if M is a complete n-manifold, satisfying (i) the volume growth condition (1.1), (ii) Liouville Theorem for harmonic functions, and either (v) a generalized Poincar & eacute;- Sobolev inequality (1.5), or (vi) a general integral inequality (1.6), and Liouville Theorem for harmonic map u : M -> K with Sec(K) <= 0, then (1) M has only one end and (2) there is no nontrivial homomorphism from fundamental group pi(1)(partial derivative D) into pi(1)(K) as stated in Theorem 1.5. Some applications in geometry (3), geometric analysis (4), nonlinear partial differential systems (5), integral inequalities on complete noncompact manifolds (6) are made (cf. e.g., Theorems 3.1, 4.1, 5.1, and 6.1). Whereas we made the first study in ([29, 32]) on how the existence of an essential positive supersolution of a second order partial differential systems Q(u) = 0 on a Riemannian manifold M, (by which we mean a C-2 function v >= 0 on M that is positive almost everywhere on M, and that satisfies Q(v) = div(A(x, v, del v)del v) + b(x, v, del v)v <= 0 (5.1) ) affects topology, geometry, analysis and variational problems on the manifold M. Whereas we generate the work in [35], under p-parabolic stable condition without assuming the p-th volume growth condition lim(r ->infinity)r(-p) Vol(B-r) = 0. The techniques, concepts, and results employed in this paper can be combined with those of essential positive supersolutions of degenerate nonlinear partial differential systems (cf. for example, Theorems 5.1- 5.5, 6.1, etc.) generalizing previous work in [32, 4.11], which in term recaptures the work of Schoen-Simon-Yau ([25, Theorem 2]). The combined techniques, concepts and method of [32] and [35] can also be used in other new manifolds we found by an extrinsic average variational method ([34]).