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数学代写|黎曼几何代写Riemannian geometry代考|Extrinsic Criteria for Weighted Parabolicity

In this section we analyze the weighted parabolicity of a submanifold $P$ immersed in a weighted manifold $M$ as it was done, (in the Riemannian case), in Sections 14 , 18 and 19 . In this case we consider the restriction to $P$ of the distance function to the pole, (the extrinsic distance), and the family of capacitors in the submanifolds, constituted by the extrinsic balls.

Our first result is an extension to the weighted setting of previous theorems for Riemannian manifolds in [EP] by Esteve and the third author, and in [MaP4, MaP7]. We note that the particular situations of weighted $(w, f)$-model spaces were established in Theorems $3.2$ and $3.3$ of [HPR1].

In [HPR2] a parabolicity criterion is established assuming lower bounds on the $q$-weighted sectional curvatures of the ambient manifold and upper bounds on the derivatives of the weight and the radial weighted mean curvatures of the submanifold.

数学代写|黎曼几何代写Riemannian geometry代考|The Grigor’yan–Fernandez Criterion

Our aim in this subsection is to give a weighted version of the hyperbolicity criterion studied in Section 19 (see also [CHS] for a weighted version with a modified isoperimetric profile).

Let $M$ be a Riemannian manifold with a continuous density $f=e^{h}$. For any open set $\Omega \subseteq M$, the weighted isoperimetric profile of $\Omega$ is the function $I_{\Omega, h}$ : $\left[0, V_{h}(\Omega)\right] \rightarrow \mathbb{R}$ defined by $I_{\Omega, h}(0)=0$, and
$I_{\Omega, h}(v)=\inf \left{A_{h}(\Sigma) ; \Sigma \subseteq \bar{\Omega}\right.$ is a compact hypersurface enclosing weighted volume $v}$,
for any $\left.v \in\left(0, V_{h} \Omega\right)\right]$
Remark 27.1. Obviously we get the weighted isoperimetric inequality
$$A_{h}(\Sigma) \geq I_{\Omega, h}\left(V_{h}(E)\right) \geq I_{M, h}\left(V_{h}(E)\right)$$
for any open set $E \subseteq \Omega$ with smooth boundary $\Sigma$.
Remark 27.2. On the other hand, if we consider the function $h=0$, and the sets $\Omega_{0}={p}$ and $\Omega=M$ in Definition $19.1$, then, given $t \geq 0$, the weighted isoperimetric profile of $\Omega$ is the $\Omega_{0}={p}$-rooted isoperimetric profile of $M$, namely, $I_{\Omega, h}(t)=\phi_{M,{p}}(t) .$

We consider $\Omega$ a precompact domain in $M$ and $\psi: \Omega \rightarrow \mathbb{R}$ a smooth function such that $\psi(\Omega)=[a, b]$ with $a<b$. Following the notation of Section 6 , applying formula (6.2) with $u=g f=g e^{h}$, and assuming that the set of critical points of $\psi$, denoted as $\Omega_{0}$, has measure zero we deduce,
\begin{aligned} \int_{\Omega} g|\nabla \psi| d V_{h} &=\int_{a}^{b}\left(\int_{\Sigma_{t}} g d A_{h, t}\right) d t \ \frac{d}{d t} V_{h}(t) &=\int_{\Gamma(t)}|\psi(x)|^{-1} d A_{h, t} \end{aligned}

where $V_{h}(t)=\operatorname{Vol}{h}(\Omega(t))$. So we can state the following weighted version of the co-area formula: Theorem 27.3. i) For every integrable function $u$ on $\bar{\Omega}$ : $$\int{\Omega} u \cdot|\nabla \psi| d V_{h}=\int_{a}^{b}\left(\int_{\Gamma(t)} u d A_{h, t}\right) d t$$
where $d A_{h, t}$ is the weighted volume element of $\Gamma(t)$.
ii) The function $V_{h}(t):=\operatorname{Vol}{\mathrm{h}}(\Omega(t))$ is a smooth function on the regular values of $\psi$, where its derivative is given by (assuming that the set of critical points of $\psi$ has measure zero): $$\frac{d}{d t} V{h}(t)=\int_{\Gamma(t)}|\nabla \psi|^{-1} d A_{h, t}$$
Remark 27.4. If $g \geq 0$ on $\Omega$ and assuming that the set of critical points of $\psi, \Omega_{0}$, has null measure, we have, using the first equation in (27.2), that
\begin{aligned} \int_{\Omega} g d V_{h} &=\int_{\Omega_{0}} g d V_{h}+\int_{\Omega-\Omega_{0}}\left(g|\nabla \psi|^{-1}\right)|\nabla \psi| d V_{h} \ &=\int_{a}^{b}\left(\int_{\Sigma_{t}} g|\nabla \psi|^{-1} d A_{h, t}\right) d t \end{aligned}
We are now ready to prove a lower bound for the capacity via the isoperimetric profile function.

数学代写|黎曼几何代写Riemannian geometry代考|Graphs and Flows

In the two final sections of these notes we now briefly indicate how many of the concepts from the previous sections can be carried over almost verbatim to locally finite graphs and thus give a fruitful alternative understanding of what is going on. We first recall that the dimension of $P^{m}$ has so far been assumed to be $m>1$, partly because the intrinsic geometry of geodesic segments is completely trivial. This viewpoint is, of course, altered considerably if we allow geodesic segments to be joined in such a way as to form a metric geodesic graph in the ambient space. The analysis of restricted distance functions on minimal geodesic graphs in Riemannian manifolds has been studied in [Ma7] following essentially the same lines of reasoning – and the same set of goals – as in the first edition of the present notes.

In view of the new results for (sub-)manifolds that we have reported above, there are then several interesting and pertinent challenges and questions that are calling for similar results to hold true on (minimal) metric graphs in (weighted) manifolds with suitable bounds on their curvatures. Indeed, the notions of Dirichlet spectrum, mean exit time moment spectrum, the weighted capacities, the type problem, etc. are all well defined on such graph-structures. Therefore the quest for finding “good” relations between the Dirichlet spectrum and the moment spectrum in that setting is quite natural. This also holds for the quest of answering the Kac question for either one of the two spectra. These questions can thus be studied following the same lines – or other refined lines with graph theoretic and other discrete tools at our disposal – as in the previous smooth geometric settings. We refer to the papers [McT)Ma, McПMMh, CKT], which contain interesting results in this direction.

Here, however, we will only consider the purely combinatorial structure of graphs, which already by itself can be considered as carrier of such fundamental notions as potential functions, capacity, isoperimetric inequalities, etc., and for which we may also state the type problem, i.e., whether random walk on the graph is transient or recurrent. This latter question will be addressed and discussed in detail via a concrete example in the next section.

We let $G=(V, E)$ denote a finite graph with edge set $E$ and vertex set $V$. If we associate the resistance value of $1(\mathrm{Ohm})$ to every edge and consider the current from one vertex $a$ to another vertex $b$ in $G$, then we are studying potential theory on finite networks. We refer to [DoyS], [So], and [Wo]. Following [DoyS] we show explicitly the relation between the capacity energy and the harmonic potential in the case of a finite graph.

数学代写|黎曼几何代写Riemannian geometry代考|The Grigor’yanFernandez Criterion

\eft 的分隔符缺失或无法识别

$$A_{h}(\Sigma) \geq I_{\Omega, h}\left(V_{h}(E)\right) \geq I_{M, h}\left(V_{h}(E)\right)$$

$$\int_{\Omega} g|\nabla \psi| d V_{h}=\int_{a}^{b}\left(\int_{\Sigma_{t}} g d A_{h, t}\right) d t \frac{d}{d t} V_{h}(t) \quad=\int_{\Gamma(t)}|\psi(x)|^{-1} d A_{h, t}$$

$$\int \Omega u \cdot|\nabla \psi| d V_{h}=\int_{a}^{b}\left(\int_{\Gamma(t)} u d A_{h, t}\right) d t$$

ii) 功能 $V_{h}(t):=\operatorname{Vol} \mathrm{h}(\Omega(t))$ 是一个关于正则值的平滑函数 $\psi$ ，其中它的导数由下式给出（假设临界点的集合 测量为零) :

$$\frac{d}{d t} V h(t)=\int_{\Gamma(t)}|\nabla \psi|^{-1} d A_{h, t}$$

$$\int_{\Omega} g d V_{h}=\int_{\Omega_{0}} g d V_{h}+\int_{\Omega-\Omega_{0}}\left(g|\nabla \psi|^{-1}\right)|\nabla \psi| d V_{h} \quad=\int_{a}^{b}\left(\int_{\Sigma_{t}} g|\nabla \psi|^{-1} d A_{h, t}\right) d t$$

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MATLAB代写

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