Ngô Quốc Anh

October 12, 2013

Bochner-type formula for the conformal Killing operator on manifolds with boundary

Filed under: Uncategorized — Tags: , — Ngô Quốc Anh @ 8:39

Given a Riemannian manifold (M,g) without boundary, in the previous note, we derived a Bochner-type formula for the conformal Killing operator \mathbb L. Precisely, we obtained

\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

or equivalenlty,

\displaystyle \frac{1}{2}\int_M |\mathbb LX |^2 dv_g=- \int_M {({g^{ij}}{\nabla _i}{\nabla _j}{X^h}){X_h} - {\rm Ric}(X,X)d{v_g}}+\left( 1-\frac{2}{n}\right)\int_M |{\rm div}X|^2 dv_g.

Today, we try to derive a similar formula for the operator \mathbb L assuming the manifold has boundary \partial M.  Our starting point again is the Bochner formula for vector fields mentioned here, i.e.

\displaystyle\frac{1}{2}\Delta (|X|^2) = |\nabla X|^2 + {\rm div}({\mathbb L_X}g)(X) - {\nabla _X}{\rm div}X - {\rm Ric}(X,X).

Using this and the formula for \Delta (|X|^2) that we derived here, we arrive at

\displaystyle\frac{1}{2}{\Delta _g}(|X{|^2}) = ({g^{ij}}{\nabla _i}{\nabla _j}{X^h}){X_h} + |\nabla X{|^2}

which now yields

\displaystyle -{\rm div}({\mathbb L_X}g)(X)=-({g^{ij}}{\nabla _i}{\nabla _j}{X^h}){X_h}- {\nabla _X}{\rm div}X - {\rm Ric}(X,X).

(more…)

Advertisements

October 11, 2013

Bochner-type formula for the conformal Killing operator on manifolds without boundary

Filed under: Uncategorized — Tags: , — Ngô Quốc Anh @ 9:59

Assuming the Riemannian manifold (M,g) is compact without boundary. In the previous post, we showed that

\displaystyle \frac{1}{2}\int_M |\mathbb L_X g|^2 dv_g= \int_M |\nabla X|^2 dv_g +\int_M |{\rm div}X|^2 dv_g - \int_M {\rm Ric}(X,X)dv_g .

Today, we use the above formula to derive a Bochner-type formula for the conformal Killing operator \mathbb L given by

\displaystyle \mathbb L X = \mathbb L_Xg - \frac{2}{n}\text{div}(X)g.

Clearly, for any vector field X,

\begin{array}{lcl} |\mathbb{L}X{|^2} &=& \displaystyle \Big|{\mathbb{L}_X}g - \frac{2}{n}( \text{div}X)g \Big|^2 \hfill \\ &=&\displaystyle {\left( {{\mathbb{L}_X}g - \frac{2}{n}( \text{div}X)g} \right)_{im}}{\left( {{\mathbb{L}_X}g - \frac{2}{n}( \text{div}X)g} \right)_{jn}}{g^{ij}}{g^{mn}} \hfill \\ &=& \displaystyle |{\mathbb{L}_X}g{|^2} + \frac{4}{{{n^2}}}{( \text{div}X)^2}\underbrace {{g_{im}}{g_{jn}}{g^{ij}}{g^{mn}}}_{\delta _n^i\delta _i^n = n} - \frac{2}{n}( \text{div}X)\left[ {{g_{im}}{\mathbb{L}_X}{g_{jn}} + {g_{jn}}{\mathbb{L}_X}{g_{im}}} \right]{g^{ij}}{g^{mn}} \hfill \\ &=& \displaystyle |{\mathbb{L}_X}g{|^2} + \frac{4}{n}{( \text{div}X)^2} - \frac{2}{n}( \text{div}X)[({\mathbb{L}_X}{g_{jn}})\underbrace {{g_{im}}{g^{ij}}{g^{mn}}}_{\delta _m^j{g^{mn}} = {g^{jn}}} + ({\mathbb{L}_X}{g_{im}})\underbrace {{g_{jn}}{g^{ij}}{g^{mn}}}_{\delta _n^i{g^{mn}} = {g^{im}}}] \hfill \\ &=&\displaystyle |{\mathbb{L}_X}g{|^2} + \frac{4}{n}{( \text{div}X)^2} - \frac{2}{n}( \text{div}X)[({\nabla _j}{X_n}+\nabla_n X_j){g^{jn}} + ({\nabla _i}{X_m}+\nabla_m X_i){g^{im}}] \hfill \\ &=& \displaystyle |{\mathbb{L}_X}g{|^2} + \frac{4}{n}{( \text{div}X)^2} - \frac{2}{n}( \text{div}X)[\underbrace {2{\nabla ^n}{X_n} + 2{\nabla ^m}{X_m}}_{4 \text{div}X}] \hfill \\ &=& \displaystyle |{\mathbb{L}_X}g{|^2} - \frac{4}{n}{( \text{div}X)^2}. \end{array}

(more…)

June 19, 2013

The Bochner formula for vector fields

Filed under: Riemannian geometry — Tags: — Ngô Quốc Anh @ 2:50

The Bochner formula for a gradient vector field is a important tool in geometric analysis which basically says that

\displaystyle\frac{1}{2}\Delta (|\nabla u{|^2}) = |{\nabla ^2}u{|^2} + \langle \nabla \Delta u,\nabla u\rangle + {\rm Ric}(\nabla u,\nabla u)

for any function f. Whenever |\nabla u| \ne 0, we have a  formula for |\nabla u| as follows

\displaystyle\frac{1}{2}\Delta (|\nabla u|) = \frac{1}{|\nabla u|} \left( \nabla u \cdot \nabla u ( \Delta u) + {\rm Ric}(\nabla u,\nabla u) + |\nabla \nabla u|^2 - \bigg| \Big\langle \nabla \nabla u, \frac{\nabla u}{|\nabla u|}\Big\rangle\bigg|\right).

Today, we discuss a variant of it called the Bochner formula for vector fields. I found this identity in a recent preprint of Li Ma [here].

Theorem. Let (M, g) be a Riemannian manifold of dimension n. Let X be a smooth vector field on M. Then we have

\displaystyle\frac{1}{2}\Delta (|X|^2) = |\nabla X|^2 + {\rm div}({\mathbb L_X}g)(X) - {\nabla _X}{\rm div}X - {\rm Ric}(X,X)

where {\mathbb L_X}g is the Lie derivative of the vector field X with respect to underlying metric g.

(more…)

Blog at WordPress.com.