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$2\mathrm{d}$ area minimizing short embeddings

If think yes, because the assumption $f^*h \leq g$ gives pointwise bounds on the derivatives of $f$ : in local charts, $|\partial_i f|^2 \leq g_{ii}$. So up to translation, you can put the image of $f$...
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Accepted

• 30.7k
1 vote
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Weitzenböck formula and comparison of norms

I am not an expert in all the delicate points of clifford algebras and spin structures, but I think the following shows the constant can't be 1 in general: let $(M,g)$ be a Riemannian manifold of ...
• 336

Harmonic functions on complete Riemannian manifolds

For the $n$-dimensional hyperbolic space it is already the space of bounded harmonic functions that is infinitely dimensional, which follows from the integral Poisson formula.
• 15.4k
1 vote

Decomposition of tensors

If what @AMath91 meant by a codazzi tensor is a symmetric tensor $\lambda \in \Gamma(S^{2}T^{*}M)$ satisfying $d^{\nabla}\lambda=0$ then I think the decompostion in the question cannot be true. Take ...
• 336
Accepted

Why are we interested in spectral gaps for Laplacian operators

A spectral gap gives information on geometry of the manifold via Cheeger's inequality, https://en.wikipedia.org/wiki/Cheeger_constant See also Buser's inequality discussed there. More directly, a ...
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