Recently, I'm reading Geometric Analysis by Peter Li. The proof of the Cheeger-Gromoll splitting theorem in his book(Theorem 4.4) is more simple than the one on Peterson's book, which avoid the application of smooth support functions, but I don't think it is rigorous enough, so I want some help. For your reference, the book can be found at https://www.cambridge.org/core/books/geometric-analysis/D0A2375D56122B91A0BA370530978248
The Laplacian comparison theorem in Peter Li's book states the conclusion in the sense of distributions and Peterson's book states the Laplacian comparison in the sense of smooth support functions. Peterson's book proves the strong maximum principle for subharmonic functions rigorously but Peter Li's book applies the strong maximum principle in Theorem 4.4 without any explanation. So I'm wondering if we can derive from $\int_M f \Delta \phi \geq 0$ for all $\phi\in C^\infty_c(M)$, $\phi \geq 0$ that $\Delta f \geq 0$ in the sense of smooth support functions (the definition is on Peterson's book, Page 281), that is,
- $f_\varepsilon(p) = f(p)$.
- $f_\varepsilon(x) \leq f(x)$ in some neighborhood of $p$.
- $\Delta f_\varepsilon(p) \geq -\varepsilon$.
If this implication is false, what kind of strong maximum principle does Peter Li apply? Can anyone give some references for this.
Here is my latest thinking. I thought it can be solved as follows.
In Cheeger-Gromoll's original proof, they use the following definition of subhamonicity. Here, we say a continuous function $f$ is subharmonic if given any connected compact region $D$ in $M$ with smooth boundary $\partial D$, one has $f \leq h$ on $D$, where $h$ is the continuous function on $D$ which is harmonic on $Int D$ with $h|_{\partial D} = f|_{\partial D}$.
This definition obviously implies the strong maximum principle as follows. If $f$ attains its maximum in the interior of $M$, namely, at $x_0\in M$. By the maximum principle for harmonic functions, $h_D \leq 0$. However, $0 = f(x_0) \leq h_D(x_0) \leq 0$. By the strong maximum principle of harmonic functions, $h_D \equiv 0$ and hence $f \equiv 0$ on $M$ by the arbitrariness of $D$.
So now the question can be reformulated as below. Given any point $x$, can we find a neighborhood $D$ of $x$ and a specific function $\phi_0\in C^\infty_c(D)$ such that $\Delta \phi_0 \leq 0$, $\phi_0 \geq 0$ on $D$ and $\phi_0(x) > 0$ ? And by density of $C_c^\infty$ in $H^2_0$, it suffices to find such a $\phi_0 \in C^2_0$ satisfying those conditions.
If so, then we can derive that $f$ is subharmonic if $\int_M f \Delta \phi \geq 0$ for all $\phi\in C^\infty_c(M), \phi \geq 0$.
The bolded question is true when $M$ is Euclidean since we can take $\phi = R^2 - |x-x_0|^2$ and $D = B_(x_0, R)$. And I think it is also true for Riemannian manifold since if we take normal coordinates centered at $x_0$ and we still let $\phi = R^2 - |x-x_0|^2$, then $\Delta_{Euclidean}\phi = -2n$ on $D$, and since $g_{ij}(x_0) = \delta_{ij}(x_0)$ and by continuity, $\Delta_g \phi \leq 0$ for sufficiently small $R$, so this is the desired function.