Connecting Poincaré Inequality with Sobolev Inequalities on Riemannian Manifolds

Abstract

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 $\operatorname{(i)}$ the volume growth condition $(1.1)$, $\operatorname{(ii)}$ Liouville Theorem for harmonic functions, and either $\operatorname{(v)}$ a generalized Poincaré- Sobolev inequality $(1.5)$, or $\operatorname{(vi)}$ a general integral inequality $(1.6)$, and Liouville Theorem for harmonic map $u : M \to K$ with $\operatorname{Sec}^K \le 0$, then $(1)$ $M$ has only one end and $(2)$ there is no nontrivial homomorphism from fundamental group $\pi_1(\partial D)$ into $\pi_1 (K)$ as stated in Theorem $1.5$. Some applications in geometry $(\S 3)$, geometric analysis $(\S 4)$, nonlinear partial differential systems $(\S 5)$, integral inequalities on complete noncompact manifolds $(\S 6)$ 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 $v \ge 0$ on $M$ that is positive almost everywhere on $M$, and that satisfies $Q(v)=\operatorname{div}(A(x,v,\nabla v)\nabla v)+b(x,v,\nabla v)v\leq 0\quad (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 $\lim _{r \to \infty} r^{-p}\operatorname{Vol}(B_r) =0$. 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]).