Estimates for the energy density of critical points of a class of conformally invariant variational problems

Lamm, Tobias; Lin, Longzhi

doi:10.1515/acv-2012-0104

Cited by 16 publications

(26 citation statements)

References 17 publications

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“…Now using (18) and by the results of Coifman-Lions-Meyer-Semmes [5] or Wente's lemma [53], we know (25) B L ∞ (Dr(p)) + ∇B L 2,1 (Dr(p)) ≤ C ∇ũ 2 L 2 (Dr(p)) . Therefore, combining (16), (22), (24) and (25) we have…”

Section: Lemma 24 (Hardy's Inequalitymentioning

confidence: 99%

Min-max minimal disks with free boundary in Riemannian manifolds

2020

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In this paper, we establish a min-max theory for constructing minimal disks with free boundary in any closed Riemannian manifold. The main result is an effective version of the partial Morse theory for minimal disks with free boundary established by Fraser. Our theory also includes as a special case the min-max theory for Plateau problem of minimal disks, which can be used to generalize the famous work by Morse-Thompkins and Shiffman on minimal surfaces in R n to the Riemannian setting.More precisely, we generalize the min-max construction of minimal surfaces using harmonic replacement introduced by Colding and Minicozzi to the free boundary setting. As a key ingredient to this construction, we show an energy convexity for weakly harmonic maps with mixed Dirichlet and free boundaries from the half unit 2-disk in R 2 into any closed Riemannian manifold, which in particular yields the uniqueness of such weakly harmonic maps. This is a free boundary analogue of the energy convexity and uniqueness for weakly harmonic maps with Dirichlet boundary on the unit 2-disk proved by Colding and Minicozzi.

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Section: Lemma 24 (Hardy's Inequalitymentioning

confidence: 99%

Min-max minimal disks with free boundary in Riemannian manifolds

2020

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“…Now we check how P transforms ω, by (7) we have P −1 dP + P −1 ωP = ω P = d * η + P −1 d * bP ∈ L (2,1) (D) (8) and…”

Section: Proof Of the Regularity Result Corollary 22mentioning

confidence: 99%

“…In section 3 we show that critical points of conformally invariant elliptic Lagrangians solve (2) and, under an added regularity assumption, we can find the frame S solving (1). Unlike the case for harmonic maps we require the theory of Rivière, namely the existence of the frame A solving (4), in order to find S. However this can still be used to re-prove a recent estimate of Lamm and Lin [8], Theorem 3.3. We remark that it might be possible to drop the added regularity and still be able to find S in this setting; either a positive or a negative answer to this question would provide further insight into these regularity problems.…”

Section: Introductionmentioning

confidence: 99%

Critical ∂̅ problems in one complex dimension and some remarks on conformally invariant variational problems in two real dimensions

Sharp¹

2013

Advances in Calculus of Variations

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We will study a linear first order system, a connection @ problem, on a vector bundle equipped with a connection, over a Riemann surface. We show optimal conditions on the connection forms which allow one to find a holomorphic frame, or in other words to prove the optimal regularity of our solution. The underlying geometric principle, discovered by Koszul-Malgrange, is classical and well known; it gives necessary and sufficient conditions for a connection to induce a holomorphic structure on a vector bundle over a complex manifold. Here we explore the limits of this statement when the connection is not smooth and our findings lead to a very short proof of the regularity of harmonic maps in two dimensions as well as re-proving a recent estimate of Lamm and Lin concerning conformally invariant variational problems in two dimensions.

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“…Besides the uniqueness consequence, this theorem is very useful for many other problems such as flow convergence, see [25]. In fact our proof simplifies the original proof by Colding-Minicozzi and the one of Lamm-Lin [22]. In fact our idea applies probably to all conformally invariant problem, since it relies only on the ε-regularity, the fact that the right hand side of the equation is orthogonal to T N .…”

Section: Step 2: Convexity For Free Boundary Energymentioning

confidence: 90%