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Index, Intersections, and Multiplicity of Min-Max Geodesics
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abstract
We prove upper bounds for the Morse index and number of intersections of min-max geodesics achieving the $p$-widths of a closed surface. A key tool in our analysis is a proof that for a generic set of metrics, the tangent cone at any vertex of any finite union of closed immersed geodesics consists of exactly two lines. We also construct examples to demonstrate that multiplicity one does not hold generically in this setting. Specifically, we construct an open set of metrics on $S^2$ for which the $p$-width is only achieved by $p$ copies of a single geodesic.
Forward citations
Cited by 3 Pith papers
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Non-persistence of strongly isolated singularities, and geometric applications
For generic metrics, stationary varifolds with only strongly isolated singularities either are smooth or have a more complicated singularity; in codimension one, only smooth or non-strongly-isolated objects persist.
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Stable $s$-minimal cones in $\mathbb{R}^2$ are flat for $s \sim 0$
The paper claims that for small values of the fractional perimeter parameter s, the only stable s-minimal cones in R^2 are half-planes, but a key integral estimate in the proof is false.
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The p-widths of $RP^2$
The p-widths of (RP^2, g_std) are ω_p = 2π floor((1+sqrt(1+8p))/4), achieved by Z2-invariant polynomial sweepouts.
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