| The present thesis investigate two geometric analysis problems:one is the fundamental gap problem of eigenvalues on convex domains in unit sphere Sn,an-other is local Sobolev constant estimate for integral Bakry-(?)mery Ricci curvature and its applications.The fundamental gap problem,namely,the optimal lower bound estimate for the fundamental gap(difference between the first eigenval-ues),is an important topic in Geometric analysis field.The famous fundamental gap conjecture(a complementary problem of §IV the spectrum in Yau’s prob-lem section[69])claim that the sharp lower bound of the fundamental gap on convex domain of Rn,for the Laplacian operator or the Schrodinger operator with convex potential under Dirichlet boundary condition,is 3π2/D2,where D is the diameter of a convex domain.Through decades of hard work of mathemati-cians,the conjecture was completely solved by B.Andrews and J.Clutterbuck in their celebrated work[2]in 2011,and they conjectured similar results hold for spaces with constant sectional curvature.We study the analogous fundamental conjecture for spherical convex domains which proves the sharp lower bound for fundamental gap estimate under the assumption that the diameter D ≤ π/2 and the the dimension n>3.The assumption conditions are cancelled in the subse-quent work[31,39]of G.Wei et.al.On the other hand,Sobolev inequality is an important tool in Geometric analysis,Sobolev constant estimate usually depends on the volume comparison,so it also closely related to the curvature of manifolds.Under different curvature conditions,local Sobolev constant estimate has a long history.One of the latest conclusion,appeared in the work of Dai-Wei-Zhang[32],is local Sobolev constant estimate for integral curvature without noncollapsing condition.The thesis extend it to the integral Bakry-(?)mery Ricci curvature case.To solve the problems mentioned above,the work of this thesis can be de-tailed as follow:In the first chapter,we review the research background and recent progress of the fundamental gap problem,as well as the development and present situation of the Sobolev inequality under different curvature conditions.Then on this foundation,the main work and innovation are introduced in this paper.In the second chapter we mainly consider the fundamental gap conjecture in unit sphere Sn,and prove that when n ≤ 3 and D,the diameter of a convex do-main in the Rn is no more than π/2 the sharp lower bound of the fundamental gap is 3π2/D2.For this purpose,we firstly study the fundamental gap of one-dimensional model space,and obtain the optimal lower bound of the gap is 3π2/D2 if n ≥ 3.Sec-ondly,we prove the key super log-concavity comparison theorem for the first eigenfunction,and obtain the condition of this property hold is the diameter of convex domain no more than π/2.Thirdly,we derive a gap comparison using the log-concavity comparison theorem and "Laplacian comparison" for two points distance function.In the third chapter,applying the equivalence of the local Sobolev constant and isoperimetric constant,we estimate the local Sobolev inequality for integral Bakry-(?)mery Ricci curvature in the complete smooth metric measure space.As an application,we obtain the maximum principle and gradient estimate under the curvature assumption. |
