Comparison of volume forms on Riemannian manifolds.
$begingroup$
I am reading the Cheng's paper (1975), which states that
Theorem. Suppose $M$ is a complete Riemannian manifold and Ricci curvature of $Mgeq(n-1)k, n=mathrm{dim} M.$ Then, for $x_0in M$ we have $$lambda_1(B(x_0,r_0))leqlambda_1(V_n(k,r_0))$$ and equality holds if and only if $B(x_0,r_0)$ is isometric to $V_n(k,r_0)$. Here $B(x_0,r_0)$ denotes the ball of radius $r_0$ with center $x_0$ in $M$, and $V_n(k,r_0)$ the ball of radius $r_0$ in the space form of constant curvature $k$, $lambda_1$ the first eigenvalue of the Laplacian on a prescribed domain.
This comparison starts from the inequality $$frac{dtheta(txi)}{dt}bigg/theta(txi)leqfrac{dtheta_k^n(txi)}{dt}bigg/theta_k^n(txi),$$ where $theta(txi)$ is understood to be $sqrt{mathrm{det}(g_{ij})}times t^{-n+1}$ with respect to the normal coordinate.
The author says that it is obtained by standard techniques on Jacobi fields, but I do not see what they are. Could anyone give me a reference for it?
Thanks.
riemannian-geometry volume
asked Dec 10 '18 at 12:15
JJWJJW
263
$endgroup$
add a comment |
$begingroup$
I am reading the Cheng's paper (1975), which states that
Theorem. Suppose $M$ is a complete Riemannian manifold and Ricci curvature of $Mgeq(n-1)k, n=mathrm{dim} M.$ Then, for $x_0in M$ we have $$lambda_1(B(x_0,r_0))leqlambda_1(V_n(k,r_0))$$ and equality holds if and only if $B(x_0,r_0)$ is isometric to $V_n(k,r_0)$. Here $B(x_0,r_0)$ denotes the ball of radius $r_0$ with center $x_0$ in $M$, and $V_n(k,r_0)$ the ball of radius $r_0$ in the space form of constant curvature $k$, $lambda_1$ the first eigenvalue of the Laplacian on a prescribed domain.
This comparison starts from the inequality $$frac{dtheta(txi)}{dt}bigg/theta(txi)leqfrac{dtheta_k^n(txi)}{dt}bigg/theta_k^n(txi),$$ where $theta(txi)$ is understood to be $sqrt{mathrm{det}(g_{ij})}times t^{-n+1}$ with respect to the normal coordinate.
The author says that it is obtained by standard techniques on Jacobi fields, but I do not see what they are. Could anyone give me a reference for it?
Thanks.
riemannian-geometry volume
asked Dec 10 '18 at 12:15
JJWJJW
263
$endgroup$
add a comment |
$begingroup$
I am reading the Cheng's paper (1975), which states that
Theorem. Suppose $M$ is a complete Riemannian manifold and Ricci curvature of $Mgeq(n-1)k, n=mathrm{dim} M.$ Then, for $x_0in M$ we have $$lambda_1(B(x_0,r_0))leqlambda_1(V_n(k,r_0))$$ and equality holds if and only if $B(x_0,r_0)$ is isometric to $V_n(k,r_0)$. Here $B(x_0,r_0)$ denotes the ball of radius $r_0$ with center $x_0$ in $M$, and $V_n(k,r_0)$ the ball of radius $r_0$ in the space form of constant curvature $k$, $lambda_1$ the first eigenvalue of the Laplacian on a prescribed domain.
This comparison starts from the inequality $$frac{dtheta(txi)}{dt}bigg/theta(txi)leqfrac{dtheta_k^n(txi)}{dt}bigg/theta_k^n(txi),$$ where $theta(txi)$ is understood to be $sqrt{mathrm{det}(g_{ij})}times t^{-n+1}$ with respect to the normal coordinate.
The author says that it is obtained by standard techniques on Jacobi fields, but I do not see what they are. Could anyone give me a reference for it?
Thanks.
riemannian-geometry volume
asked Dec 10 '18 at 12:15
JJWJJW
263
$endgroup$
I am reading the Cheng's paper (1975), which states that
Theorem. Suppose $M$ is a complete Riemannian manifold and Ricci curvature of $Mgeq(n-1)k, n=mathrm{dim} M.$ Then, for $x_0in M$ we have $$lambda_1(B(x_0,r_0))leqlambda_1(V_n(k,r_0))$$ and equality holds if and only if $B(x_0,r_0)$ is isometric to $V_n(k,r_0)$. Here $B(x_0,r_0)$ denotes the ball of radius $r_0$ with center $x_0$ in $M$, and $V_n(k,r_0)$ the ball of radius $r_0$ in the space form of constant curvature $k$, $lambda_1$ the first eigenvalue of the Laplacian on a prescribed domain.
This comparison starts from the inequality $$frac{dtheta(txi)}{dt}bigg/theta(txi)leqfrac{dtheta_k^n(txi)}{dt}bigg/theta_k^n(txi),$$ where $theta(txi)$ is understood to be $sqrt{mathrm{det}(g_{ij})}times t^{-n+1}$ with respect to the normal coordinate.
The author says that it is obtained by standard techniques on Jacobi fields, but I do not see what they are. Could anyone give me a reference for it?
Thanks.
riemannian-geometry volume
riemannian-geometry volume
asked Dec 10 '18 at 12:15
JJWJJW
263
asked Dec 10 '18 at 12:15
JJWJJW
263
asked Dec 10 '18 at 12:15
JJWJJW
263
asked Dec 10 '18 at 12:15
JJWJJW
263
asked Dec 10 '18 at 12:15
JJWJJW
263
263
add a comment |
add a comment |
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