Degree counting for Toda system with simple singularity: One point blow up

Lee, Youngae; Lin, Chang–Shou; Yang, Wen; Zhang, Lei

doi:10.1016/j.jde.2019.09.016

Cited by 10 publications

(7 citation statements)

References 52 publications

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“…We also mention here some related problems, including Toda systems and mean field equations, see Chen-Lin [5,6] and references therein. In particular the recent result by Lee-Lin-Yang-Zhang [13] considering the singularity formation of two-points collision is morally related to the cone points merging behavior studied in this paper.…”

Section: Review Of Existing Resultsmentioning

confidence: 64%

Conical metrics on Riemann surfaces, I: The compactified configuration space and regularity

Mazzeo

Zhu

2020

Geom. Topol.

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We introduce a compactification of the space of simple positive divisors on a Riemann surface, as well as a compactification of the universal family of punctured surfaces above this space. These are real manifolds with corners. We then study the space of constant curvature metrics on this Riemann surface with prescribed conical singularities at these divisors. Our interest here is in the local deformation for these metrics, and in particular the behavior as conic points coalesce. We prove a sharp regularity theorem for this phenomenon in the regime where these metrics are known to exist. This setting will be used in a subsequent paper to study the space of spherical conic metrics with large cone angles, where the existence theory is still incomplete.

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Section: Review Of Existing Resultsmentioning

confidence: 64%

Conical metrics on Riemann surfaces, I: The compactified configuration space and regularity

Mazzeo

Zhu

2020

Geom. Topol.

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“…we have, by estimates for single equations, 22) it is easy to see that the harmonic function that eliminates the oscilla-…”

Section: This Estimate Of V Kmentioning

confidence: 96%

“…This means for 4πm < ρ 1 < 4π(m + 1), 4πN < ρ 2 < 4π(N + 1), there should be a topological degree that only depends on m, N and the genus of the manifold M. On the other hand, it is crucial to understand the asymptotic behavior of bubbling solutions when (ρ 1 , ρ 2 ) tends to the grid points (4πm, 4πn). A lot of work on blowup analysis and degree counting has been done when one of ρ k i s crosses a multiple of 4π while the other stays away from 4πN (see [21,22] for example). In this article we initiate a direct attack to the study of bubbling solutions when both parameters are tending to critical positions.…”

Section: Introductionmentioning

confidence: 99%

Estimates of bubbling solutions of $SU(3)$ Toda systems at critical parameters-Part 1

Wu¹,

Zhang²

2019

Preprint

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For regular SU(3) Toda systems defined on Riemann surface, we initiate the study of bubbling solutions if parameters (ρ k 1 , ρ k 2 ) are both tending to critical positions:) or (4πN, 4π) (N > 0 is an integer). We prove that there are at most three formations of bubbling profiles, and for each formation we identify leading terms of ρ k 1 − 4π and ρ k 2 − 4πN, locations of blowup points and comparison of bubbling heights with sharp precision. The results of this article will be used as substantial tools for a number of degree counting theorems, critical point at infinity theory in the future.

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“…Our main goal is to prove the local uniqueness of blow up solutions of (1.3) with collapsing singularities. We remark that the study of blow up solutions of (1.3) with collapsing singularities is also important to compute the topological degree for the Toda system as noticed in [24,26], where the degree counting of the whole system is reduced to computing the degree of the corresponding shadow system (see [24,Theorem 1.4]). Thus it is inevitable to encounter with the phenomena of collapsing singularities when we want to find a priori bound for solutions of an associated shadow system.…”

Section: Introductionmentioning

confidence: 99%

Uniqueness of bubbling solutions with collapsing singularities

Lee

Lin

2019

Journal of Functional Analysis

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The seminal work [7] by Brezis and Merle showed that the bubbling solutions of the mean field equation have the property of mass concentration. Recently, Lin and Tarantello in [31] found that the "bubbling implies mass concentration" phenomena might not hold if there is a collapse of singularities. Furthermore, a sharp estimate [23] for the bubbling solutions has been obtained. In this paper, we prove that there exists at most one sequence of bubbling solutions if the collapsing singularity occurs. The main difficulty comes from that after re-scaling, the difference of two solutions locally converges to an element in the kernel space of the linearized operator. It is well-known that the kernel space is three dimensional. So the main technical ingredient of the proof is to show that the limit after re-scaling is orthogonal to the kernel space.Date: September 28, 2018. 1 2 YOUNGAE LEE AND CHANG-SHOU LIN However, there is a big difference when the collapsing singularities are considered. First, Lin and Tarantello in [31] observed a new phenomena such that blow-up solutions with collapsing singularities might not have the "concentration" property of its mass distribution. The general version was studied in [23]. To describe the results, let us consider the following equation:where lim t→0 q i (t) = q / ∈ {q d+1 , · · · , q N }, ∀ i = 1, · · · , d and q i (t) = q j (t) for i = j ∈ {1, · · · , d}. Then the following holds:t be a sequence of blow up solutions of (1.3) with ρ / ∈ 8πN. Then u * t blows up only at q ∈ M. Furthermore, there exists a function w * such that. So Theorem B tells us that the mass concentration does no longer hold if the collapsing singularity occurs. Indeed, we have lim t→0 M h * e u * t dv g < +∞, which is different from the situation described in Theorem A. We note that Theorem B could be improved provided that the following nondegeneracy condition holds: Definition 1.1. A solution w * of (1.4) is said non-degenerate, if the linearized problem

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Degree counting for Toda system with simple singularity: One point blow up

Cited by 10 publications

References 52 publications

Conical metrics on Riemann surfaces, I: The compactified configuration space and regularity

Conical metrics on Riemann surfaces, I: The compactified configuration space and regularity

Estimates of bubbling solutions of $SU(3)$ Toda systems at critical parameters-Part 1

Uniqueness of bubbling solutions with collapsing singularities

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