Existence of Harmonic Maps with Two-Form and Scalar Potentials
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Existence of Harmonic Maps with Two-Form and Scalar Potentials Xiangzhi Cao1,2 · Qun Chen1 Received: 23 February 2020 / Accepted: 25 May 2020 / © Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2020
Abstract In this paper, we obtain existence results of Dirichlet problem for harmonic maps with two-form and scalar potentials from Riemannian surfaces M with nonempty boundary into compact manifolds N with convex boundary and complete manifolds N via the heat flow method. Keywords Dirichlet problem · Heat flow · Harmonic map · Two-form · Potential Mathematics Subject Classification (2010) 58E20 · 35K55 · 53C08 · 53C80
1 Introduction Let (M, g) be a compact Riemannian surface with boundary, (N, h) a Riemannian manifold, u ∈ C 2 (M, N ). Consider the following functional: 1 |du|2 + u∗ B + H (u) , (1.1) E(u) = M 2 where H ∈ C 2 (N ) is function and B is a two-form on N . This is a geometric variational model which generalizes the usual harmonic maps from Riemannian surfaces. In physics literatures, the full action for the bosonic string is of this type [7]. We recall that anyelliptic functional which is conformal invariant can be written as (c.f. [3]): E(u) = 1 2 + u∗ B , the regularity of its critical points was studied in [8]. |du| M 2 The Euler–Lagrange equation of the functional (1.1) is (1.2) τ (u) − Z du(e1 ) ∧ du(e2 ) − ∇H (u) = 0, Dedicated to J¨urgen Jost on the occasion of his 65th birthday. Qun Chen
[email protected] Xiangzhi Cao [email protected] 1
Hubei Key Laboratory of Computational Science, Wuhan University, Wuhan, 430072, China
2
School of Mathematical Science, Sichuan Normal University, Chengdu, 610068, China
X. Cao, Q. Chen
where τ (u) is the tension field of u, {e1 , e2 } is local orthonormal frame on M, Z ∈ (Hom(2 T ∗ N, T N )) and B, Z satisfy dB(U, V , W ) = Z(U ∧ V ), W ,
dB = ,
for U, V , W ∈ (T N ) and is a three-form on N . A map satisfying (1.2) is called harmonic map with two-form and scalar potentials. The heat flow for (1.2) is given by ∂u = τ (u) − Z du(e1 ) ∧ du(e2 ) − ∇H (u) on M × [0, Tmax ). (1.3) ∂t From the Nash’s embedding theorem, N can be embedded into some Euclidean space by i : N → RK . Then (1.2) can be written as
du(e1 ) ∧ du(e2 ) − ∇H (u) = 0, u − A(u)(du, du) − Z and the corresponding heat flow is ∂u
du(e1 ) ∧ du(e2 ) − ∇H (u), = u − A(u)(du, du) − Z ∂t
and ∇H are extensions of Z and ∇H to RK respectively (c.f. [2, 6]), which in where Z the sequel we still denote by Z and ∇H , respectively. A(u)(·, ·) is the second fundamental form of N in RK . The purpose of this paper is to derive existence results for the Dirichlet problem of such maps. The case that M is a compact Riemannian surface without boundary was considered in [1]. We will prove our results via considering the following initial-boundary value problem: ⎧ ∂u on M × [0, Tmax ); ⎨ ∂t = τ (u) − Z(du(e1 ) ∧ du(e2 )) − ∇H (u) (1.4) u = u0 on ∂M × [0, Tmax ); ⎩ on M × {0}, u = u0 where u0 ∈ C 2+α (M, N ) for some α ∈ (0, 1).
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