In this thesis we study three problems in the field of geometric analysis: eigenvalue estimate of non-linear operators, existence of minimal surfaces and isoperimetric problems. These problems are more or less related to the topic of geometric calculus of variations, which is the study of extreme points of functionals defined on manifolds.

The first part is devoted to the study of lower bound of the principal eigenvalue of a family of non-linear elliptic operator $L_p$. Using the gradient and maximum comparison technique developed in \cite{Ko18} together with ideas from \cite{LW19eigenvalue2}, we proved that on a compact metric measure space(possibly with convex boundary) $(M,g,m)$ with curvature-dimension condition $BE(\kappa, N) (\kappa \neq 0)$, if $L$ is a elliptic diffusion operator whose invariant measure is $m$, then the principal eigenvalue of $L_p$ is bounded below by the first eigenvalue of a one-dimensional ODE with Neumann boundary condition. We showed that this is sharp result by constructing examples of metric measure space $M$ on which the eigenvalue problem of $L_p$ degenerates into the model equation problem. This work extends the $\kappa = 0$ case proved in \cite{Ko18}.

The second part is devoted to the study of existence of free boundary minimal hypersurfaces in compact manifolds, from a min-max theoretical point of view. Following the ideas from \cite{ambrozio2018min} and \cite{marques2019equidistribution}, we prove that in a simply connected compact manifold $(M,\partial M, g)$ under certain conditions) with its metric that is locally maximising the width of $M$, there is a sequence of equidistributed free boundary minimal hypersurfaces.

The third part is devoted to the study of anisotropic isoperimetric inequality for regions outside of a ball in $\mathbb{R}^n$. Based on Alexandrov-Bakelman-Pucci's Method, we use the concept of generalized normal cone introduced by \cite{LWW20}, to show that for any region outside a Euclidean ball, its isoperimetric ratio has a lower bound that can only be achieved by a half-Wulff shape cut by a half-space.

Ricci flow is a powerful and fundamentally innovative tool in the field of geometric analysis introduced by Richard Hamilton [Ha82] in 1982. Many longstanding geometric and topological problems have been solved using Ricci flow. For example, the Poincaré conjecture, Thurston’s geometrization conjecture, the $1/4$-pinched differentiable sphere theorem, the generalized Smale conjecture, and so on. In the seminal works of Hamilton and Perelman, it is crucial to understand at least the qualitative behavior of the singularities so that Ricci flow can be continued via surgeries. To understand the singularity formation, it is desirable to classify singularity models, which are the blow-up limits along sequences of space-time points with curvature tending to infinity, or at least to understand them qualitatively well enough for topological applications. However, as compared to dimension 3, the geometry becomes drastically more complicated starting from dimension 4, and so does the singularity analysis of Ricci flow. Recently, Richard Bamler in [Bam20a,Bam20b,Bam20c] established a groundbreaking theory for the weak limits (his F-limits) of Ricci flows on closed manifolds in higher dimensions. This theory will be fundamentally important in the study of higher-dimensional Ricci flow singularities. All the known examples so far suggest that Ricci flow singularity models should be mostly shrinking or steady Ricci solitons. These are the self-similar ancient solutions to Ricci flow, and they can be viewed as generalized Einstein manifolds. Thus, it is vitally important to study shrinking and steady Ricci solitons that arise as singularity models.

In the dissertation, we shall survey some recent results on the geometry in the large of singularity models or more general noncollapsed ancient solutions of Ricci flow, which were jointly investigated by collaborators and the author. We shall streamline some proofs and also present some new unpublished findings. There are two fundamental notions of space-time blow-downs for ancient flows: asymptotic shrinking solitons by Perelman [Per02] and tangent flows at infinity by Bamler [Bam20c]. We will show that the two notions coincide, which is the main theorem of [CMZ21a]. We will present that for ancient flows, various entropy quantities introduced mainly by Perelman converge to those of the tangent flows at infinity, which were proved in [MZ21,CMZ21a,CMZ21b,CMZ21d]. It will also be demonstrated that how tangent flows at infinity determine the geometry in the large of steady solitons in dimension 4, which is the main theorem of [BCDMZ]. Moreover, we will present an optimal volume growth estimate for noncollapsed steady solitons in all dimensions, which is the main result of [BCMZ21].

This thesis is a summary of the work accomplished by the author and his coauthors in geometric analysis during his Ph.D. studies. It consists of four parts.

The first and the second parts are the estimates of modulus of continuity for viscosity solutions of nonlinear partial differential equations in domains in Euclidean spaces and on manifolds. The main results generalize B. Andrews and J. Clutterbuck's modulus of continuity estimates for smooth solutions to viscosity solutions. The main ingredients of the proofs are the parabolic maximum principle for semicontinuous functions, its generalized version on manifolds, and the multi-point estimates method.

The third part studies asymptotic behavior of nonparametric hypersurfaces of dimension $n$

moving by $\a $ powers of its Gaussian Curvature with $\a >1/n$.

Our work generalizes the results for $\a =1$ obtained by V. Oliker to all $\a > 1/n$.

Although we are using similar ideas, the proof is quite technical.

In the fourth part, we study classification of shrinking gradient Ricci solitons.

Our main result asserts that any four-dimensional complete gradient shrinking Ricci soliton with positive isotropic curvature

is either a quotient of $\mathbb{S}^4$ or a quotient of $\mathbb{S}^3 \times \R $.

This gives a clean classification result removing the earlier additional assumptions in by L. Ni and N. Wallach.

%An immediate corollary is that any four-dimensional gradient shrinking soliton with positive curvature operator must be isometric to $S^4$.

This also generalizes a result of Perelman on three-dimensional gradient shrinking Ricci solitons to dimension four.

The result has important consequences in studying Ricci flow on four-dimensional manifolds.

In this dissertation we will first give an exposition for some topics on Perelman's entropy and some results related to the analysis of the entropy, and then present the content of two papers among the author's publication list.

Chapter 1 is an introduction to Perelman's entropy and the author's main results. The statement of these main results can be found in section 1.3.

Chapter 2 and chapter 3 are expository materials on Perelman's entropy and its related analytic tools; these results are included because of their importance to our main theorems.

In chapter 4, we prove some estimates for the Nash entropy on ancient solutions and thereby prove a gap theorem for the asymptotic entropy.

In chapter 5, we prove an assertion made by Perelman, saying that for an ancient solution to the Ricci flow with bounded and nonnegative curvature operator, bounded entropy is equivalent to $\kappa$-noncollapsing on all scales. This proof is based on accurate gaussian upper and lower estimates for the conjugate heat kernel.