- Why is a² + b² ≥ 2ab ?
- A remark on the divergence theorem
- The Cauchy-Schwarz inequality and the Lagrange identity
- On the existence of a metric compatible with a given connection
- A curious identity on the median triangle
- 27 lines on a smooth cubic surface
- Weighted Hsiung-Minkowski formulas and rigidity of umbilic hypersurfaces
- The discrete Gauss-Bonnet theorem
- Why a vector field rotates about its curl?
- A functional inequality on the boundary of static manifolds
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- Mean value properties for harmonic functions on Riemannian manifolds
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Category Archives: Differential equations
Question: Suppose we are given a torsion-free (i.e. the torsion tensor vanishes) affine connection on a smooth connected manifold . Does there exist a Riemannian metric such that its Levi-Civita connection is ? If so, is it unique if we … Continue reading
1. A motivating example Consider the -Laplace equation in , where is a nonempty bounded open subset of , and on . The energy functional associated with the PDE is . For , because of convexity, has a unique minimizer … Continue reading
The Burgers equation is the following: Here can be regarded as the (scalar) velocity of a fluid and can be regarded as the viscosity. This equation can be regarded as the simplified version of the Euler’s equation without the pressure … Continue reading
This is a sequel to my previous post Reilly type formula and its applications. (Oops… it’s been a long time!) 1. Introduction Integral formulas have always been an important tool for studying various analytical and geometric problems on Riemannian manifolds. … Continue reading