As I've learned more Riemannian geometry, many of my teachers have said that studying the Laplacian (and its eigenvalues) is very important. But I must admit, I've never fully understood why.
Fundamentally, I would like to know why the Laplacian is important among all differential operators on a Riemannian manifold. I would also like to know what geometric information the Laplacian is supposed to encode.
That being said, I have spent a little time thinking about all this, and my current understanding is as follows:
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I've heard that the Laplacian is the "simplest" isometrically-invariant "scalar differential operator" on a Riemannian manifold. If true, this statement would convince me of its importance. However, I don't know to what extent this is true.
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An isometric immersion $f \colon S \to M$ is harmonic iff it is a minimal submanifold of $M$. In particular, an isometrically immersed submanifold of $\mathbb{R}^n$ is minimal iff its coordinate functions are harmonic.
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The Euler-Lagrange equation for the Dirichlet energy is $\Delta f = 0$. (But why we care about minimzing energy is also somewhat mysterious to me.)
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Weitzenböck formulas comparing two elliptic second-order differential operators (and especially Laplacians) give Bochner-type vanishing theorems.
I should point out that I'm aware that harmonic functions satisfy many of the nice properties that complex-analytic functions do (by virtue of elliptic regularity and maximum principle magic). Still, this doesn't quite tell me why I should care about the Laplace operator itself.
Note: I'm aware of this related question on the eigenvalues of the Laplacian. But again, my interest is in Riemannian geometry; matters of applied mathematics (while interesting) are not my focus right now.
Best Answer
There is one important piece of evidence that has not been mentioned in the "comments" above, namely, de Rham's theorem for compact manifolds, and the idea that harmonic forms uniquely represent (real) cohomology classes. That certainly testifies to the significance of the Laplacian (the kernel of which consists of harmonic forms).