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数学代写|黎曼几何代写Riemannian geometry代考|Weighted Rotationally Symmetric Spaces

As already alluded to in Section 17, the works [GreW, Gri1, O’N], provide a complete description of the rotationally symmetric model spaces $M_{w}^{m}$, that we will use to establish our comparison theorems. We remark that these spaces constitute a huge set of comparison spaces that we will now apply in the same way as we used the constant curvature spaces $\mathbb{K}^{n}(b)$ in Section 3. The spaces $\mathbb{K}^{n}(b)$ are but particular examples of rotationally symmetric model spaces. We shall need the formal definition of warped products as follows:

Definition $23.1$ (see [GreW, Ch. 2], [Gri1, Sect. 3], [Pe, Ch. 3]). A $w$-model space is a smooth warped product $\left(M_{w}^{m}, g_{w}\right):=B^{1} \times_{w} F^{m-1}$ with base $B^{1}:=[0, \Lambda[\subset \mathbb{R}$ (where $0<\Lambda \leq \infty$ ), fiber $F^{m-1}:=\mathbb{S}{1}^{m-1}$ (the unit $(m-1)$-sphere with standard metric), and warping function $w:[0, \Lambda[\rightarrow[0, \infty[$ such that $w(r)>0$ for all $r>0$, whereas $w(0)=0, w^{\prime}(0)=1$, and $w^{(k)}(0)=0$ for all even derivation orders $k$. The point $o{w}:=\pi^{-1}(0)$, where $\pi$ denotes the projection onto $B^{1}$, is called the center point of the model space. If $\Lambda=\infty$, then $o_{w}$ is a pole of the manifold (recall that a pole of a complete Riemannian manifold $M$ is a point $o \in M$ such that the exponential map $\exp {o}: T{o} M \rightarrow M$ is a diffeomorphism).

Example 23.2. The simply connected space forms $\mathbb{K}^{m}(b)$ of constant sectional curvature $b$ can be constructed as $w$-models with any given point as center point using the warping functions
$$w_{b}(r):= \begin{cases}\frac{1}{\sqrt{b}} \sin (\sqrt{b} r) & \text { if } b>0 \ r & \text { if } b=0 \ \frac{1}{\sqrt{-b}} \sinh (\sqrt{-b} r) & \text { if } b<0\end{cases}$$ Note that, for $b>0$, the function $w_{b}(r)$ admits a smooth extension to $r=\pi / \sqrt{b}$. For $b \leq 0$ any center point is a pole.

In a model space the sectional curvatures for 2-planes containing the radial direction from the center point are determined by the radial function
$$-\frac{w^{\prime \prime}(r)}{w(r)}$$
where $r=$ dist $_{M_{w}}\left(o_{w}, \cdot\right)$ is the distance from the center $o_{w} \in M_{w}$. Moreover, the mean curvature of the metric sphere of radius $r$ from the center $o_{w}$ is
$$\frac{w^{\prime}(r)}{w(r)}=\frac{d}{d r} \ln (w(r))$$

数学代写|黎曼几何代写Riemannian geometry代考|Weighted Curvatures

Let us consider a complete non-compact weighted manifold $\left(N^{n}, g, e^{h}\right)$. Among the several generalizations of the Ricci curvature tensor (which we use to control the Laplacian), the most extensively used are the Bakry-Emery Ricci tensors:

• The finite or $q$-Bakry-Emery Ricci tensor, $q>0$ :
$$\operatorname{Ric}_{h}^{q}=\text { Ric }-\text { Hess } h-\frac{1}{q} d h \otimes d h .$$
• The infinite or $\infty$-Bakry-Emery Ricci tensor:
$$\operatorname{Ric}_{h}=\mathrm{Ric} \quad \mathrm{Hess} h .$$

Observe that
$$\mathrm{Ric}{h}=\mathrm{Ric}{h}^{q}+\frac{1}{q} d h \otimes d h,$$
so lower bounds on $\mathrm{Ric}{h}^{q}$ implies lower bounds on $\mathrm{Ric}{h}$. Here Ric denotes the Ricci tensor in $(N, g)$.

The weighted radial sectional curvatures, (which agree, up to a constant, with the weighted sectional curvatures introduced in [W]), are defined as follows:

Definition 24.1. Consider the radial plane $\sigma_{p} \subseteq T_{p} N$ spanned by linearly independent unit vectors $X$ and $Y$, i.e., $\sigma_{p}=\operatorname{span}{X, Y}$. We define the $\infty$-weighted sectional curvature of $\sigma_{p}$ in a weighted Riemannian manifold $\left(N^{n}, g, e^{h}\right)$ as
$$\operatorname{Sec}{h}^{\infty}\left(\sigma{p}\right)=K_{N}\left(\sigma_{p}\right)-\frac{1}{n-1}(\operatorname{Hess} h){p}(Y, Y) \text {. }$$ The $q$-weighted sectional curvature of the radial plane $\sigma{p}=\operatorname{span}{X, Y}$ is defined by:
$$\operatorname{Sec}{h}^{q}\left(\sigma{p}\right)=K_{N}\left(\sigma_{p}\right)-\frac{1}{n-1}(\operatorname{Hess} h){p}(X, Y)-\frac{1}{(n-1) q}(d h \otimes d h)(Y, Y)$$ In the paper [HPR2] an analysis of the intrinsic distance function to a pole in the weighted manifold $M^{m}$ is carried out. Analogously, as in Sections 3 and 8 , the estimates of the Hessian and the weighted Laplacian of the distance from the pole lead to capacity comparison results and parabolicity criteria for weighted manifolds under lower bounds on the Ricci curvatures $\mathrm{Ric}{h}$ and $\mathrm{Ric}_{h}^{q}$.

We also found in [HPR2] bounds for the weighted isoperimetric quotients and volumes of metric balls, along the lines of Section 8 , but now from an intrinsic point of view. We must remark that the works [Q, L, Mo2, WW, PRRS] contain previous comparison results that involve weighted volumes and quotients of weighted volumes (although not weighted isoperimetric quotients) when Ric ${ }_{\infty}^{h}$ is bounded from below.

数学代写|黎曼几何代写Riemannian geometry代考|Analysis of Restricted Distance Functions

In [HPR1] and [HPR2] the restricted distance to a submanifold immersed in an ambient weighted manifold, is analyzed in the same vein as in Section 3. Let $P^{m}$ be an $m$-dimensional submanifold with $\partial P=\emptyset$ properly immersed in a weighted manifold $\left(N^{n}, g, \epsilon^{h}\right)$ with a pole $o \in N$. We consider in $P$ the induced Riemannian metric. We use the notation $\nabla^{P} u$ and $\Delta^{P} u$ for the gradient and Laplacian in $P$ of a function $u \in C^{2}(P)$.

The restriction to $P$ of the weight $e^{h}$ in $N$ produces a structure of weighted manifold in $P$. From (21.2) the associated $h$-Laplacian $\Delta_{h}^{P}$ has the expression
$$\Delta_{h}^{P} u=\Delta^{P} u+\left\langle\nabla^{P} h, \nabla^{P} u\right\rangle,$$ for any $u \in C^{2}(P)$. We say that the submanifold $P$ is $h$-parabolic when $P$ is weighted parabolic considered as a weighted manifold. Otherwise we say that $P$ is $h$-hyperbolic. By Theorem $22.1$ the $h$-parabolicity of $P$ is equivalent to $\operatorname{Cap}{h}^{P}(D)=$ 0 for some precompact open set $D \subseteq P$, where $\operatorname{Cap}{h}^{P}$ denotes the $h$-capacity relative to $P$. Clearly a compact submanifold $P$ is $h$-parabolic.

Now we present another necessary ingredient to establish these results: the weighted mean curvature of submanifolds. In the case of two-sided hypersurfaces this was first introduced by Gromov [G], see also [Ba, Ch. 3].

Definition 25.1. The weighted mean curvature vector or h-mean curvature vector of $P$ is the vector field normal to $P$ given by
$$\bar{H}{h}^{P}:=m \bar{H}^{P}-(\nabla h)^{\prime},$$ where $(\nabla h)^{\perp}$ is the normal projection of $\nabla h$ with respect to $P$ and $\bar{H}^{P}$ is the mean curvature vector of $P$. This is defined as $m \bar{H}^{P}:=-\sum{i=1}^{n-m}\left(\operatorname{div}^{P} N_{i}\right) N_{i}$, where $\operatorname{div}^{P}$ stands for the divergence relative to $P$ and $\left{N_{1}, \ldots, N_{n-m}\right}$ is any local orthonormal basis of vector fields normal to $P$.

We say that $P$ has constant $h$-mean curvature if $\left|\bar{H}{h}^{P}\right|$ is constant on $P$. If $\bar{H}{h}^{P}=0$, then $P$ is called $h$-minimal. More generally, $P$ has bounded h-mean curvature if $\left|\bar{H}{h}^{P}\right| \leq c$ on $P$ for some constant $c>0$. For later use we note that the equality $$\left\langle m \bar{H}^{P}, \nabla r\right\rangle+\left\langle\nabla^{P} h, \nabla^{P} r\right\rangle=\left\langle\bar{H}{h}^{P}, \nabla r\right\rangle+\langle\nabla h, \nabla r\rangle$$
holds on $P-{o}$. This easily follows from the definition of $\bar{H}_{h}^{P}$ and the fact that $\nabla h-(\nabla h)^{\perp}=\nabla^{P} h$.

As in Proposition 3.13, some inequalities for the weighted Laplacian of submanifolds have been established under bounds (which are not necessarily constants) on the radial sectional curvatures of the ambient manifold. In the particular case of a weighted $(w, f)$-model space it can be checked that all the estimates in the next statement become equalities.

−在′′(r)在(r)

数学代写|黎曼几何代写Riemannian geometry代考|Weighted Curvatures

• 有限的或q-Bakry-Emery Ricci 张量，q>0 :
里克Hq= 里克 − 赫斯 H−1qdH⊗dH.
• 无限或∞-Bakry-Emery Ricci 张量：
里克H=里克赫斯H.

数学代写|黎曼几何代写Riemannian geometry代考|Analysis of Restricted Distance Functions

ΔH磷在=Δ磷在+⟨∇磷H,∇磷在⟩,对于任何在∈C2(磷). 我们说子流形磷是H-抛物线当磷是加权抛物线，被视为加权流形。否则我们说磷是H-双曲线。按定理22.1这H- 抛物线磷相当于帽H磷(D)=0 对于一些预紧开集D⊆磷， 在哪里帽H磷表示H-容量相对于磷. 显然是一个紧凑的子流形磷是H-抛物线。

H¯H磷:=米H¯磷−(∇H)′,在哪里(∇H)⊥是的正常投影∇H关于磷和H¯磷是平均曲率向量磷. 这被定义为米H¯磷:=−∑一世=1n−米(div磷ñ一世)ñ一世， 在哪里div磷代表相对于的散度磷和\left 的分隔符缺失或无法识别\left 的分隔符缺失或无法识别是向量场的任何局部正交基，垂直于磷.