The Sobolev embedding inequality on manifolds

Question

Let $(M,g)$ be a (smooth) compact Riemanian manifold of dimension $n$. I expect that the following inequality is true for any smooth function $f$:

$$(\int_{M} |f|^{\beta})^{1/\beta} \leq C \;(\int_{M} |\nabla f|_{g}^{2}+\int_{M} f^{2})^{1/2},$$ for $\beta=2n/(n-2)$. Since this is true for $M=\mathbb{R}^{n}$.

Could anyone point out a clear reference for this?

In the book: Aubin T. Nonlinear analysis on manifolds: Monge Ampere equations, page 50, the author stated this result with the extra assumption: $M$ has a constant curvature and positive injectivity radius! Are these conditions necessary?

Answer 1

I found the result you referenced in page 50 of the following text:

Aubin, Thierry, Some nonlinear problems in Riemannian geometry, Springer Monographs in Mathematics. Berlin: Springer. xviii, 396 p. (1998). ZBL0896.53003.

I assume is the book you are looking at. The result of compact manifolds without boundary is proved in Theorem 2.20 the same text, and a slightly more general result is proved on page 45 which asserts the following.

2.21 Theorem. The Sobolev imbedding theorem holds for $M_n$ a complete manifold with bounded curvature and injectivity radius $\delta>0.$ Moroever, for any $\varepsilon>0,$ there exists a constant $A_q(\varepsilon)$ such that every $\varphi \in H_q^q(M_n)$ satisfies $$ \lVert \varphi \rVert_p \leq \left[K(n,q)+\varepsilon\right] \lVert \nabla \varphi \rVert_q + A_q(\varepsilon) \lVert \varphi \rVert_q,$$ with $1/p=1/q-1/n>0,$ and $K(n,q)$ is the optimal constant for Sobolev embedding in $\Bbb R^n.$

Note that if $M$ is smooth and compact, the injectivity radius is always positive and the curvature is always bounded so this applies. The result in page 50 (Theorem 2.28) shows that if $M$ is compact and either $n=2$ or $M$ is flat, one can additionally take $\varepsilon=0$ in the above.

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In general the main obstruction to the Sobolev embedding result is the degeneration of the metric; you can think of having $M = \Bbb R^n$ with a metric $g(x) = w(x)^{\frac12} g_0$ with $g_0$ the usual Euclidean metric, and $w>0$ a suitable weight function, the inequality reduces to $$ \left( \int_M |f|^{\beta} \,\mathrm{d} x \right)^{\frac1\beta} \leq C \left( \int_M w(x)|\nabla f|^2 \,\mathrm{d} x + \int_{\Bbb R^n} |f|^2 \,\mathrm{d}x\right)^{\frac12}.$$ From here we observe the validity of this inequality depends on choice of weight function $w.$ If $M$ is compact this cannot occur however, and a straightforward partition of unity argument given in Theorem 2.20 will do the trick.

In the case of the results in Aubin, one is interested in obtaining sharp bounds for the associated constants, which has implications on the geometry of the space $M.$ For this one needs the more refined arguments given in the book and the papers cited therein, involving the exponential map and using curvature bounds in addition.