Nonlinear Calderón–Zygmund inequalities for maps

Abstract

Being motivated by the problem of deducing $\mathsf {L}^{p}$-bounds on the second fundamental form of an isometric immersion from $\mathsf {L}^{p}$-bounds on its mean curvature vector field, we prove a nonlinear Calderón–Zygmund inequality for maps between complete (possibly noncompact) Riemannian manifolds.

Appendix A. On the infinite harmonic radius

This appendix is devoted to a proof of the following

Proposition

Let (M, g) be a complete, noncompact, m-dimensional Riemannian manifold and assume that there exists some $o \in M$ and some $\alpha \in (0,1)$ such that $r_{1,\alpha }(o)= +\infty$. Then, (M, g) is isometric to the Euclidean $\mathbb {R}^m$.

Proof

By assumption, there exist a sequence of rays $R_k \rightarrow +\infty$ and a corresponding sequence $\varphi _k\,{:}\,B_{R_k}(o) \rightarrow \mathbb {R}^m$ of harmonic coordinates charts centered at o such that conditions (2) and (3) are satisfied. We consider the corresponding sequence $(B_{R_k}(o),g,o)$ of pointed Riemannian manifolds, and we show that it has a subsequence that converges in the $\mathsf {C}^{1}$-topology to $(\mathbb {R}^m,g_\infty ,0)$, where $g_\infty$ is a scalar product with constant coefficients. In particular, $(\mathbb {R}^m,g_\infty ,0)$ is isometric to $\mathbb {R}^m$. Since the same subsequence of pointed Riemannian manifolds obviously converges in the $\mathsf {C}^{\infty }$-topology to (M, g, o), we obtain the desired conclusion. Indeed, pointed $\mathsf {C}^{1}$-convergence implies pointed GH convergence and, since the limit metric spaces are proper (they are complete Riemannian manifolds), they must be metrically isometric. But metrically isometric Riemannian manifolds are isometric in the Riemannian sense by the Myers–Steenrod theorem.

Let

$$\begin{aligned} \Omega _k = \varphi _k(B_{R_k}(o)), \quad g_k = (\varphi _{k}^{-1})^*g, \quad 0_k = \varphi _k(o)=0. \end{aligned}$$

Observe that, by condition (2), $B^{\mathbb {R}^{m}}_{R_{k}/\sqrt{2}} (0)\Subset \Omega _k$. It follows that $\{\Omega _k\}$ exhausts $\mathbb {R}^m$. Moreover, $\Omega _k \Subset B^{\mathbb {R}^{m}}_{\sqrt{2}R_{k}} (0) \Subset \Omega _{k^\prime }$, for every $k^\prime \gg 1$. In particular, each $\Omega _k$ is relatively compact and, up to extracting a subsequence, we can assume that $\Omega _k \Subset \Omega _{k+1}$. Now, fix $k_0$. According to property (3), for every $k \gg 1$ we have that the metric coefficients $(g_k)_{ij}$ of $g_k$ are uniformly bounded in $\mathsf {C}^{1,\alpha }({\bar{\Omega }}_{k_0})$. Since, by Ascoli-Arzelá, for any domain $D \Subset \mathbb {R}^m$ and any $0< \beta < \alpha$ the embedding $\mathsf {C}^{1,\alpha }(D) \hookrightarrow \mathsf {C}^{1,\beta }(D)$ is compact, we deduce that a subsequence $(g_{k'})_{ij}$ converges in the $\mathsf {C}^{1}$-topology on $\Omega _{k_0}$. Now, we let $k_0$ increase to $+\infty$ and we use a diagonal argument to deduce the existence of a $\mathsf {C}^1$ metric $g_\infty$ on $\mathbb {R}^m$ such that, in the coordinates of $\mathbb {R}^m$, a suitable subsequence $\{g_{k''}\}$ $\mathsf {C}^1$-converges to $g_\infty$ uniformly on compact sets. We stress that $g_\infty$ is actually Riemannian and it is bi-Lip equivalent to the Euclidean metric $g_E$. Indeed, by taking the limit in condition (2) along $g_{k''}$ gives that $2^{-1}\cdot g_{\mathbb {R}^{m}} \le g_\infty \le 2 \cdot g_{\mathbb {R}^{n}}$. We show that $g_{\infty }$ has constant coefficients. To this end we recall that, by condition (3), for every $k'' > k''_0$ it holds

$$\begin{aligned} \sup _{\Omega _{k^{\prime \prime }_0}} |\partial (g_{k^{\prime \prime }})_{ij}| \le \frac{1}{R_{k^{\prime \prime }}}. \end{aligned}$$

Letting $k^{\prime \prime } \rightarrow +\infty$ shows that the metric coefficients $(g_\infty )_{ij}$ are constant on $\Omega _{k^{\prime \prime }_0}$. Since $k^{\prime \prime }_0$ is arbitrary, the claimed property follows. In particular, the covariant differentiation with respect to $g_\infty$ is Euclidean.

In conclusion, we have obtained that $(\Omega _{k^{\prime \prime }},g_{k^{\prime \prime }},0_{k^{\prime \prime }})$ $\mathsf {C}^1$-converges to $(\mathbb {R}^m,g_\infty ,0)$ and this, by definition, means that $(B_{k^{\prime \prime }}(o), g, 0)$ $\mathsf {C}^1$-converges to the same pointed Riemannian manifold. $\square$