Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Eigenvalues of the drifted Laplacian on complete metric measure spaces   I In this paper, first we study a complete smooth metric measure space
$(M^n,g, e^{-f}dv)$ with the ($\infty$)-Bakry-\'Emery Ricci curvature
$\textrm{Ric}_f\ge \frac a2g$ for some positive constant $a$. It is known that
the spectrum of the drifted Laplacian $\Delta_f$ for $M$ is discrete and the
first nonzero eigenvalue of $\Delta_f$ has lower bound $\frac a2$. We prove
that if the lower bound $\frac a2$ is achieved with multiplicity $k\geq 1$,
then $k\leq n$, $M$ is isometric to $\Sigma^{n-k}\times \mathbb{R}^k$ for some
complete $(n-k)$-dimensional manifold $\Sigma$ and by passing an isometry,
$(M^n,g, e^{-f}dv)$ must split off a gradient shrinking Ricci soliton
$(\mathbb{R}^k, g_{can}, \frac{a}{4}|t|^2)$, $t\in \mathbb{R}^k$. This result
has an application to gradient shrinking Ricci solitons. Secondly, we study the
drifted Laplacian $\mathcal{L}$ for properly immersed self-shrinkers in the
Euclidean space $\mathbb{R}^{n+p}$, $p\geq1$ and show the discreteness of the
spectrum of $\mathcal{L}$ and a logarithmic Sobolev inequality.

8
The categories of this paper are: hep-ex
21
The categories of this paper are: adap-org
9
The categories of this paper are: nlin
19
The categories of this paper are: eess
6
The categories of this paper are: astro-ph
0
The categories of this paper are: hep-ph
30
The categories of this paper are: q-alg
16
The categories of this paper are: stat
35
The categories of this paper are: mtrl-th
1
The categories of this paper are: cs
28
The categories of this paper are: alg-geom
37
The categories of this paper are: supr-con
20
The categories of this paper are: acc-phys
11
The categories of this paper are: hep-lat
10
The categories of this paper are: q-bio
31
The categories of this paper are: ao-sci