# Homogeneous regular manifolds

In order to solve the well-known Plateau-Problem on a general (non-compact) Riemannian 3-manifold, Morrey first introduced the condition of homogeneous regularity and defined it in the following way:

1. A riemannian 3-manifold $(M,g)$ is homogeneously regular if there exist positive constants $k,K$ such that every point $p \in M$ lies in the image of a chart $\psi$ with domain the unit Ball $B_1(0) \subset \mathbb R^3$ such that for all $x \in B_1(0)$ and all $v \in \mathbb R^3$, we have $k||v||^2 \leq g_{ij}(\psi(x))v_i v_j \leq K||v||^2$ (Here ||.|| denotes the standart eucledian norm).

Later, Meeks, Simon and Yau reformulated this condition in the following way:

1. A riemannian 3-manifold $(M,g)$ is homogeneously regular if there exist positive constants $\nu,\mu,\alpha$ such that for every $p \in M$, the exponential map $exp_p: B_\alpha(0) \subset T_pM \to G_\alpha(p)$ provides a diffeomorphism between the eucledian ball of radius $\alpha$ and the geodesic ball in $M$ about $p$ of the same radius. Further, it is required that $|d exp_p|, |d exp_p^{-1}| \leq \nu$ and the corresponding metric tensor $g_{ij}$ satisfies $|\frac{\partial g_{ij}}{\partial x_k}| \leq \frac{\mu}{\alpha}$ and $|\frac{\partial^2g_{ij}}{\partial x_k\partial x_l}| \leq \frac{\mu}{\alpha^2}$ on all of $B_\alpha(0)$.

It is also claimed (but not explicitly proven) that, using comparsion theorems, one can show that both conditions are equivalent to:

1. $(M,g)$ has injectivity radius greater than $0$ and bounded sectional curvature.

Condition 2. certainly implies the lower bound of the injectivity radius in 3. Moreover, since the sectional curvature is a rational expression in the partial derivaties of the metric tensor up to order 2, it should be possible to prove the boundedness of the sectional curvature just from that observation. However, I have no idea how to prove that 3 implies 2. There are many comparsion theorems in differential geometry, and I don't know which one would be of most use here. Also, I do not see how 1 implies either 2 or 3. Can anyone help me here ?

• It seems that 1 is weaker than 3 and 3 is weaker than 2. Sep 21, 2020 at 14:16