2014 / June

New Comparison Theorems In Riemannian Geometry

Yingbo Han, Ye Li, Yibin Ren, Shihshu Walter Wei

radial curvature; Hessian; Laplacian; Jacobi equation; Riccatti equation, radial curvature; Hessian; Laplacian; Jacobi equation; Riccatti equation

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163-186

We construct and use solutions, subsolutions, and supersolutions of differential equations as catalysts to link hypotheses on radial {\it curvature} of a complete $n$-manifold $(M,g)$ to conclusions on the analysis or geometry of ${\it quadratic}$ ${\it forms}$ and second order ${\it differential}$ ${\it operators}$. These conclusions are formulated in terms of pointwise estimates on the Hessian and pointwise and weak estimates on the Laplacian of the distance function $r$ from a fixed point $x_0$ in $M\, .$ In particular, we prove Hessian Comparison Theorems and Laplacian Comparison Theorems, generalizing the work of Greene and Wu [2]: If the radial curvature $K$ of $M$ satisfies $-\frac {a^2}{c^2+r^2}\leq K(r)\leq \frac {b^2}{c^2+r^2}$ on $D(x_0)\, $ where $0 \le a^2\, , 0 \le b^2 \le \frac14\, ,$ $0 \le c^2\, ,$ and $D(x_0) = M \backslash (\operatorname{Cut}(x_0) \cup \{ x_0 \})\, ,$ then

$ \frac{1+\sqrt{1-4b^2}}{2r}\bigg(g-dr\otimes dr\bigg) \le \operatorname{Hess} (r)\leq \frac{1+\sqrt{1+4a^2}}{2r}\bigg( g-dr\otimes dr\bigg)$

on $D(x_0)\, ,$ in the sense of quadratic forms, and

$\qquad \qquad (n-1)\frac{1+\sqrt{1-4b^2}}{2r} \le \Delta r \leq (n-1)\frac{1+\sqrt{1+4a^2}}{2r}\, $

holds pointwise on $D(x_0)\, ,$ and $\Delta r \leq (n-1)\frac{1+\sqrt{1+4a^2}}{2r}\, $ holds weakly on $M\, .$

This is equivalent to that if the radial curvature $K$ on $D(x_0)\, $ satisfies

$- \frac {A(A-1)}{r^2}\leq K(r)\leq \frac {B(1-B)}{r^2}$ where $1 \le A\, ,$ and $\frac 12 \le B \le 1\, , $ then

$\frac{B}{r}\bigg( g-dr\otimes dr\bigg) \le \operatorname{Hess} (r)\leq \frac{A}{r}\bigg( g-dr\otimes dr\bigg)\, \operatorname{and}\, (n-1)\frac{B}{r}\le \Delta r \le (n-1)\frac{A}{r}$

holds pointwise on $D(x_0)\, ,$ and $\Delta r \le (n-1)\frac{A}{r}$ holds weakly on $M\, .$ We also prove and apply Hessian Comparison Theorems via Jacobi Type inequalities, Comparison Theorems on Riccati type inequalities, and Sturm Comparison Theorems. An analog of a theorem of Greene-Wu on negatively pinched manifolds, $-\frac {A}{r^2}\leq K(r)\leq -\frac {A_1}{r^2} < 0\, $ for ${\it pointwise}$ Hessian estimates is given. On positively pinched manifolds, $0 < \frac {b_1^2}{r^2}\leq K(r)\leq \frac {{b}^2}{r^2}\, ,$ pointwise Hessian estimates are also made. Pointwise Laplacian Comparison Theorems on $D(x_0)\, $ are then immediately obtained by taking traces in Hessian Comparison Theorems. The corresponding weak upper bound estimates of the Laplacian on all of $M$ are then obtained by Green's Identity and a ${\it double}$ ${\it limiting}$ argument(cf. Lemma 9.1, [PRS], [WL]).

26D15, 53C21, 53C20

2013-08-06

2013-10-13