Bishop-Gromov Inequality

The Bishop-Gromov inequality is a comparison theorem in Riemannian geometry . It is a key statement in the proof of Gromov's compactness theorem ^[1] .

Inequality is named after Richard Bishop and Mikhail Gromov .

Content

Wording

Let be $M$ ${\ displaystyle M}$ Is a complete n- dimensional Riemannian manifold with Ricci curvature bounded below

\mathrm {Ric} \geqslant (n-1)K

{\ displaystyle \ mathrm {Ric} \ geqslant (n-1) K}

for constant $K\in \mathbb {R}$ ${\ displaystyle K \ in \ mathbb {R}}$ .

Denote by $B(p,r)_{M}$ ${\ displaystyle B (p, r) _ {M}}$ a ball of radius r around a point p defined with respect to the Riemannian distance function .

Let be $\mathbb {M} ^{n}(K)$ ${\ displaystyle \ mathbb {M} ^ {n} (K)}$ denotes an n- dimensional model space. I.e $\mathbb {M} ^{n}(K)$ ${\ displaystyle \ mathbb {M} ^ {n} (K)}$ - the full n- dimensional simply connected space of constant sectional curvature $K$ ${\ displaystyle K}$ . In this way,

$\mathbb {M} ^{n}(K)$ ${\ displaystyle \ mathbb {M} ^ {n} (K)}$ is an n- sphere of radius $1/{\sqrt {K}}$ {\ displaystyle 1 / {\ sqrt {K}}} , if a $K>0$ ${\ displaystyle K> 0}$ , or
n- dimensional Euclidean space if $K=0$ ${\ displaystyle K = 0}$ , or
Lobachevsky space with curvature $K<0$ ${\ displaystyle K <0}$ .

Then for any $p\in M$ ${\ displaystyle p \ in M}$ and ${\tilde {p}}\in \mathbb {M} ^{n}(K)$ ${\ displaystyle {\ tilde {p}} \ in \ mathbb {M} ^ {n} (K)}$ function

\phi (r)={\frac {\operatorname {Vol} B(p,r)_{M}}{\operatorname {Vol} B({\tilde {p}},r)_{\mathbb {M} ^{n}(K)}}}

{\ displaystyle \ phi (r) = {\ frac {\ operatorname {Vol} B (p, r) _ {M}} {\ operatorname {Vol} B ({\ tilde {p}}, r) _ {\ mathbb {M} ^ {n} (K)}}}}

does not increase in the interval $(0,\infty )$ ${\ displaystyle (0, \ infty)}$ .

Remarks

At $K=0$ ${\ displaystyle K = 0}$ inequality can be written as follows
$\operatorname {Vol} B(p,\lambda \cdot r)_{M}\leqslant \lambda ^{n}\cdot \operatorname {Vol} B(p,r)_{M}$ ${\ displaystyle \ operatorname {Vol} B (p, \ lambda \ cdot r) _ {M} \ leqslant \ lambda ^ {n} \ cdot \ operatorname {Vol} B (p, r) _ {M}}$

at

\lambda \geqslant 1

{\ displaystyle \ lambda \ geqslant 1}

.

If r tends to zero, then the ratio approaches unity, so together with monotonicity, this means that
$\operatorname {Vol} B(p,r)_{M}\leqslant \operatorname {Vol} B({\tilde {p}},r)_{\mathbb {M} ^{n}(K)}.$ ${\ displaystyle \ operatorname {Vol} B (p, r) _ {M} \ leqslant \ operatorname {Vol} B ({\ tilde {p}}, r) _ {\ mathbb {M} ^ {n} (K )}.}$

This version was first proved by Bishop ^[2] ^[3] .

Notes

↑ Burago Yu. D. , Zalgaller V.A. , Introduction to Riemannian geometry 1991, p. 320, (22.5)
↑ Bishop, R. A relation between volume, mean curvature, and diameter. Amer. Math. Soc. Not. 10 (1963), p. 364.
↑ Bishop RL, Crittenden RJ Geometry of manifolds, Corollary 4, p. 256

[1] Burago Yu. D. , Zalgaller V.A. , Introduction to Riemannian geometry 1991, p. 320, (22.5)

[2] Bishop, R. A relation between volume, mean curvature, and diameter. Amer. Math. Soc. Not. 10 (1963), p. 364.

[3] Bishop RL, Crittenden RJ Geometry of manifolds, Corollary 4, p. 256

Bishop-Gromov Inequality

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