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.

Since the Bochner-type formula for the conformal Killing operator on manifolds without boundary, i.e.

\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

(more…)

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