Connecting Poincaré Inequality with Sobolev Inequalities on Riemannian Manifolds

We connect the Poincaré 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 Poincaré- Sobolev inequality $(1.5)$, or (vi) a general integral inequality $(1.6)$, and Liouville Theorem for harmonic map u : M → K with SecK ≤ 0, then $(1)$ M has only one end and $(2)$ there is no nontrivial homomorphism from fundamental group π₁(∂ D) into π₁ (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 C² function v ≥ 0 on M that is positive almost everywhere on M, and that satisfies Q(v)=div(A(x,v,∇ v)∇ v)+b(x,v,∇ 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 → ∞ r⁻ᵖVol(Bᵣ) =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]).