Ngô Quốc Anh

October 29, 2013

The mean curvature under conformal changes of Riemannian metrics: A global approach

Filed under: Riemannian geometry — Tags: — Ngô Quốc Anh @ 22:46

In a previous post, I showed how the mean curvature changes under a conformal change using slightly local coordinates approach. Today, I want to reconsider that topic using global approach.

As usual, suppose (M,g) is an n-dimensional Riemannian manifold with boundary \partial M. We also assume that N is an outward unit normal vector field along the boundary \partial M. By an unit normal vector field we mean g(N,N)=1 and g(N,X)=0 for any tangent vectors X of \partial M. Note that \partial M is just a hypersurface of M and we also use g to denote the induced metric of g onto \partial M. Then we have the so-called second fundamental form \mathrm{I\!I} associated to \partial M defined to be

\displaystyle \mathrm{I\!I} (X,Y)=g(X,\nabla_Y N)

for any tangent vectors X,Y of \partial M.

TangentSpace

Regarding to the mean curvature H, we shall use the following definition

\displaystyle H=\text{trace}_g (\mathrm{I\!I})=g^{ij}\mathrm{I\!I}(\partial_i, \partial_j)=g^{ij} g(\partial_i,\nabla_{\partial_j} N).

Our aim is to calculate the mean curvature H under the following conformal change \widehat g=\phi^\kappa g for some smooth positive function \phi and a real number \kappa. It is important to note that by \widehat g=\phi^\kappa g we mean

\displaystyle \widehat g_{ij} = \phi^\kappa g_{ij}.

Therefore, for the inverse metric, there holds

\displaystyle \widehat g^{ij} = \phi^{-\kappa} g^{ij}.

Since the normal vector field N depends on the metric g, it turns out that under the new metric \widehat g,

\displaystyle \widehat N = \phi^{-\kappa/2}N

is a new normal vector field along \partial M, i.e. \widehat g(\widehat N, \widehat N)=1. To continue, we need to understand the conformal change for the Levi-Civita connection, let us recall the following formula

\displaystyle {{\widehat\nabla }_X}Y - {\nabla _X}Y = \frac{\kappa }{2}\left( {\frac{{{\nabla _X}\phi }}{\phi }Y + \frac{{{\nabla _Y}\phi }}{\phi }X - g(X,Y)\frac{{\nabla \phi }}{\phi }} \right).

According to the note, the formula we quoted should be

\displaystyle {\widehat\nabla _X}Y = {\nabla _X}Y + X(f)Y + Y(f)X - g(X,Y) {\rm grad}f

provided \widehat g=e^{2f}g. However, in view of our notation used, there holds f = \frac{\kappa }{2}\log \phi. Hence

\displaystyle {\rm grad}f=\frac{\kappa }{2}\frac{\nabla\phi}{\phi}, \quad X(f)=\nabla_Xf=\frac{\kappa }{2}\frac{\nabla_X\phi}{\phi},\quad Y(f)=\frac{\kappa }{2}\frac{\nabla_Y \phi}{\phi}.

Then we calculate the new second fundamental form given by

\displaystyle \widehat{\mathrm{I\!I}} (X,Y)=\widehat g(X,\widehat\nabla_Y \widehat N)

Clearly, we have

\begin{array}{lcl} \widehat{\mathrm{I\!I}} (X,Y) &=& \displaystyle\widehat g(X,{{\widehat\nabla }_Y}\widehat N) \hfill \\ &=& \displaystyle {\phi ^\kappa }g(X,{{\widehat\nabla }_Y}({\phi ^{ - \kappa /2}}N)) \hfill \\ &=& \displaystyle {\phi ^\kappa }g \Big( X,{\phi ^{ - \kappa /2}}{{\widehat\nabla }_Y}N - \frac{\kappa }{2}{\phi ^{ - \kappa /2 - 1}}N{{\widehat\nabla }_Y}\phi \Big) \hfill \\ &=& \displaystyle {\phi ^{\kappa /2}}g \bigg( X,{{\widehat\nabla }_Y}N - \underbrace {\frac{\kappa }{2}N\frac{{{{\widehat\nabla }_Y}\phi }}{\phi }}_{\-0} \bigg) \hfill \\ &=& \displaystyle {\phi ^{\kappa /2}}g \Bigg( X,\underbrace {{\nabla _Y}N}_{\-\mathrm{I\!I}} + \frac{\kappa }{2}\bigg( {\underbrace {\frac{{{\nabla _Y}\phi }}{\phi }N}_{\-0} + \frac{{{\nabla _N}\phi }}{\phi }Y - \underbrace {g(Y,N)}_{ = 0}\frac{{\nabla \phi }}{\phi }} \bigg) \Bigg) \hfill \\ &=& \displaystyle {\phi ^{\kappa /2}}\mathrm{I\!I} (X,Y) + {\phi ^{\kappa /2}}g \Big( X,\frac{\kappa }{2}\frac{{{\nabla _N}\phi }}{\phi }Y \Big). \end{array}

Thus, we have shown that

\displaystyle \widehat{\mathrm{I\!I}} (X,Y)= {\phi ^{\kappa /2}}\left( {\mathrm{I\!I} (X,Y) + \frac{\kappa }{2}\frac{{{\nabla _N}\phi }}{\phi }g(X,Y)} \right).

Then

\begin{array}{lcl} {\widehat H_{\widehat g}} &=& \displaystyle {{\text{trace}}_{\widehat g}}(\widehat {\mathrm{I\!I}}) \hfill \\ &=&\displaystyle {{\widehat g}^{ij}}\widehat {\mathrm{I\!I}}({\partial _i},{\partial _j}) \hfill \\ &=&\displaystyle {\phi ^{ - \kappa }}{g^{ij}}\left( {{\phi ^{\kappa /2}}\mathrm{I\!I} ({\partial _i},{\partial _j}) + {\phi ^{\kappa /2}}g \Big( {\partial _i},\frac{\kappa }{2}\frac{{{\nabla _N}\phi }}{\phi }{\partial _j} \Big) } \right) \hfill \\ &=&\displaystyle {\phi ^{ - \kappa /2}}\bigg( {{H_g} + \frac{\kappa }{2}\frac{{{\nabla _N}\phi }}{\phi }\underbrace {{g^{ij}}{g_{ij}}}_{n - 1}} \bigg). \end{array}

It is important to remind that the metric g appearing in the above calculation is just the induced metric on \partial M which is of dimension n-1, therefore g^{ij}g_{ij}=n-1 is clear. In particular, when \kappa=4/(n-2), we obtain

\displaystyle {\widehat H_{\widehat g}} = {\phi ^{ - 2/(n - 2)}} \bigg( {H_g} + \frac{{2(n - 1)}}{{n - 2}}\frac{{{\nabla _N}\phi }}{\phi } \bigg).

Therefore, if we normalize the mean curvature by

\displaystyle H_g =\frac{1}{n-1}\text{trace}_g (\mathrm{I\!I})

we then have the following rule

\displaystyle {\widehat H_{\widehat g}} = {\phi ^{ - 2/(n - 2)}} \bigg( {H_g} + \frac{2}{{n - 2}}\frac{{{\nabla _N}\phi }}{\phi } \bigg).

We now consider the case when the mean curvature H_g is defined to be

\displaystyle H_g=g^{ij} g(\partial_i,\nabla_{\partial_j} (-N)).

Clearly, H_g=-\text{trace}_g(\mathrm{I\!I}). Therefore, we can write

\displaystyle \widehat H_{\widehat g}=-\text{trace}_{\widehat g} (\widehat{\mathrm{I\!I}}) = -{\phi ^{ - 2/(n - 2)}} \bigg( -H_g + \frac{{2(n - 1)}}{{n - 2}}\frac{{{\nabla _N}\phi }}{\phi } \bigg).

Thus, we have shown that

\displaystyle \nabla_N \phi - \frac{n-2}{2(n-1)}H_g \phi= -\widehat H_{\widehat g} \phi^{n/(n-2)}.

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