Gravitating vortices with positive curvature

Kähler-Yang-Mills Equations and Vortices

“…An existence theorem for the gravitating vortex equations (3.1) with c > 0 for a triple P 1 , L, ϕ with ϕ ̸ = 0, similar to Theorem 3.7, is obtained combining Theorem 3.5 with the converse direction in this situation, proved by Garcia-Fernandez-Pingali-Yao [21].…”

Twisted and Singular gravitating vortices

“…We divide the singularity set S into seven groups according to the Venn diagram of the three sets A = Z, B = C and C = P. The analytical assumption (A) is satisfied if and only if the the numerical assumption (N) about any singular point x is satisfied for x in the corresponding part of the Venn diagram (Figure 1), for instance if x ∈ (Z ∩ C)\P saying x = p k = q k then we should have 4ατ n k + 2(1 − β k ) < 2. We expect there is a stability explanation to these numerical assumptions as what happens in the smooth case [3,11]. Theorem 5.2 (Existence of singular cosmic strings).…”

“…To derive useful a priori estimates, we mainly use the PDE system (3.2) in this section. Let Φ = |φ| 2 h as above, then there is a C 0 bound on Φ as in [11,Lemma 4.1].…”

“…The existence of smooth solution with c = 0 on a compact surface was obtained by Yisong Yang [23] under an assumption of the multiplicities of the zeros of φ, which was later interpreted as a classical GIT (geometric invariant theory) stability condition for the P SL(2, C) action on S N P 1 and proved to be necessary including the case c > 0. This stability condition was recently proved to be sufficient for the existence of smooth solution in the case c > 0 in [11]. The case of negative c on compact Riemann surfaces of genus g ≥ 1 for a suitable range of the coupling constant was studied by [3].…”

“…The existence and uniqueness (Theorem 2.7) of twisted vortices is established in section 2, generalizing the existence and uniqueness of vortices [5,17,16]. Inspired by the continuity method used in [3,11], we study a continuity path deforming the coupling constant α while fixing the twisting forms in section 3. The openness is established when g(Σ) ≥ 2 and the twisting forms have suitable signs; the closedness is proved by generalizing the a priori estimates in [3], and therefore the existence of twisted gravitating vortex is established in this setting.…”