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Results tagged with curvature
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user 16877
Gaussian curvature, mean curvature, sectional curvature, scalar curvature, curvature tensors (Riemann, Ricci, Weyl)
3
votes
0
answers
145
views
Growth of norm of curvature under direct sum or existence of universal connection
For a Hermitian vector bundle E over a Riemannian $(X,g)$ and a unitary connection A we may define the sup norm of the curvature by:
$$\|R_A\|=\sup \{\| tracefree (R_A (v))\|_{op} \mid v \in \Lambda ^2 … Specifically we want $\cal{A}$ a connection on $\mathcal {E}$, with constant curvature, s.t. a connection $A$ on $E$ minimizing the norm, is gauge equivalent to $(f^* \mathcal {E}, f^* \mathcal A)$, for …
- 491
2
votes
2
answers
280
views
Mean curvature upper bounds and area, or geodesic curvature upper bounds and length
Let $S$ be the set of (immersed) class $A$ surfaces in $(M,g)$ with mean curvature bounded from above by a fixed constant $C$. … A preliminary version of the question would be to ask if this holds for loops in $M$, with mean curvature replaced by geodesic curvature, again assuming $\pi_1(M)$ is non-torsion. …
- 491
0
votes
Accepted
Mean curvature upper bounds and area, or geodesic curvature upper bounds and length
Indeed it is no for spheres of all dimension assuming one can find an immersed class $A$ $S^n$ with mean curvature less than a given $C(A)$. …
- 491
0
votes
2
answers
337
views
Diameter of immersed surfaces with bounded from above mean curvature
Claim:
Consider the set $S$ of closed immersed Riemann surfaces $\Sigma \subset (X,g)$ (I am particularly interested in spheres), with the magnitude of the mean curvature bounded from above by some $C> … edit 2: $C$ should be upper bound for magnitude of the mean curvature. …
- 491