# Ngô Quốc Anh

## October 20, 2013

### Rayleigh-Type Quotient For The Conformal Killing Operator on manifolds with boundary

Filed under: Uncategorized — Tags: — Ngô Quốc Anh @ 22:50

Following the previous note, today we discuss a similar Rayleigh-type quotient for the conformal Killing operator $\mathbb L$ on manifolds $(M,g)$ with boundary. We also prove that

whenever $M$ admits no non-zero conformal Killing vector fields, the following holds

$\displaystyle\inf \frac{{{{\left( {\int_M {|\mathbb LX|^2 d{v_g}} } \right)}^{1/2}}}}{{{{\left( {\int_M {|X|^{2n/(n - 2)}d{v_g}} } \right)}^{(n - 2)/(2n)}}}} > 0$

where the infimum is taken over all smooth vector fields $X$ on $M$ with $X \not\equiv 0$.

$\displaystyle \frac{1}{2}\int_M |\mathbb L X|^2 dv_g= \int_M |\nabla X|^2 dv_g + \left( 1-\frac{2}{n}\right)\int_M |{\rm div}X|^2 dv_g - \int_M {\rm Ric}(X,X)dv_g,$

is no longer available, we use a new approach in order to estimate $\int_M |\mathbb L X|^2 dv_g$ from below. To this purpose, we make use of a Riemannian version for the Korn inequality recently proved by S. Dain [here].

First, in view of Corollary 1.2 in Dain’s paper, the following inequality holds

$\displaystyle \int_M |\nabla X|^2 dv_g \leqslant C \left( \int_M |X|^2 dv_g + \int_M |\mathbb LX|^2 dv_g \right)$

for some positive constant $C$ independent of $X$. This helps us to conclude that

$\displaystyle C\int_M |\mathbb LX|^2 dv_g \geqslant \|X\|_{H^1}^2 - (C+1)\|X\|_{L^2}^2$

as in Dahl et al’ paper. Therefore, we can argue by contradiction by assuming that there exists a sequence of vector fields $\{X_k\}_k \in H^1(M)$ such that