A global weak solution to the full bosonic string heat flow

We prove the existence of a unique global weak solution to the full bosonic string heat flow from closed Riemannian surfaces to an arbitrary target under smallness conditions on the two-form and the scalar potential. The solution is smooth with the exception of finitely many singular points. Finally, we discuss the convergence of the heat flow and obtain a new existence result for critical points of the full bosonic string action.


Introduction and Results
The action functional for the full bosonic string is an important model in contemporary theoretical physics. It is defined for a map from a two-dimensional domain taking values in a manifold. The action functional consists of three contributions: Besides the Polyakov action one considers the so-called B-field action and a Dilaton contribution. For the physics background of the full bosonic string we refer to [11, p. 108]. This article is a sequel to previous work concerning the existence of critical points of the full bosonic string action. In [2] an existence result was given in the case of the domain being a closed Riemannian surface and the target a Riemannian manifold having negative sectional curvature. Moreover, a second existence result has been established in [3] for the domain being twodimensional Minkowski space and the target an arbitrary closed Riemannian manifold. The aim of this article is to extend the existence result from [2] to arbitrary targets without posing any curvature assumption. In addition, we prove a regularity result for weak solutions of the critical points of the full bosonic string action. Let us explain the geometric setup in more detail. Throughout this article (M, h) is a closed Riemannian surface and (N, g) a closed, oriented Riemannian manifold of dimension dim N ≥ 3. For a map φ : M → N we consider the square of its differential giving rise to the well-known Dirichlet energy, whose critical points are harmonic maps. Let B be a two-form on N , which we pull back by the map φ and V : N → R be a scalar function. In the physics literature the full action for the bosonic string is given by We explicitly state the dependence of the action functional on the metric of the domain M since the scalar potential V (φ) is not invariant under conformal transformations. Note that in the physics literature the scalar potential V (φ) often gets multiplied with the scalar curvature of the domain. In the mathematics literature there have been several articles dealing with energy functionals similar to (1.1). On the one hand there is the notion of harmonic maps with potential introduced in [7,8], which are critical points of (1.1) with B = 0. On the other hand there have been several studies of the heatflow of (1.1) with V = 0, see for example [16,17] and [4]. For more references on the mathematical background see the introduction of [2] and references therein. The tools that we use in this article mostly originate from the theory of harmonic maps, see [13] and the book [14] for a detailed presentation.
First of all, we analyze the regularity of weak solutions of (1.2) and prove the following The major part of this article is devoted to the study of the L 2 -gradient flow of the functional (1.1), which is given by the following evolution equation This is a natural generalization of the harmonic map heat flow from surfaces. Although most of the analytical results obtained in this article follow the ideas from the standard harmonic map heat flow we will encounter several new phenomena due to the presence of the scalar potential V (φ) in the action functional. For simplicity we will mostly assume the scalar potential is smooth. However, we will point out the influence of a potential of lower regularity on the solution of (1.3) at several places. We will prove the following Theorem 1.2. Let (M, h) be a closed Riemannian surface and (N, g) a closed Riemannian manifold. Moreover, suppose that |B| L ∞ < 1 2 and that either V (φ) ∈ C ∞ (N, R) itself is sufficiently small or that after a suitable conformal transformation on the domain the integral over the scalar potential V (φ) is sufficiently small. Then for any initial data φ 0 ∈ W 1,2 (M, N ) there exists a global weak solution , which is smooth away from at most finitely many singular points The weak solution constructed here is unique and the energy functional (1.1) is decreasing with respect to time. Moreover, there exists a sequence t k → ∞ such that φ(·, t k ) converges weakly in W 1,2 (M, N ) to a solution of (1.2) denoted by φ ∞ as k → ∞ suitably and strongly away from finitely many (1) In the case of the standard harmonic map heat flow from surfaces to general targets one can blow up the singular points that form along the flow. This procedures makes use of the fact that the harmonic map heat flow is invariant under parabolic rescaling. The inclusion of the scalar potential in the action functional (1.1) breaks the conformal invariance, as a consequence the critical points of (1.1) do not scale nicely. Hence, we cannot expect to blow up the singular points that form along (1.3).
(2) If we compare the results obtained in this article with the main results from [2] we can make the following observations: In [2] an existence result for (1.2) could be obtained under the assumption that the target manifold has negative curvature. In this article we do not impose any curvature condition on the target instead we have to make strong assumption on the scalar potential V (φ).
This article is organized as follows: In section 2 we study the regularity of weak solutions to (2.1). Afterwards, in section 3, we study the heat flow associated to (2.1) and prove Theorem 1.2. Whenever employing local coordinates, we will use Greek indices for coordinates on the domain and Latin indices for coordinates in the target. In addition, we will make use of the usual summation convention, that is we will sum over repeated indices.

Analytic aspects of the full bosonic string
In this section we want to analyze several analytical properties of solutions of (1.2). To this end we make use of the Nash embedding theorem and assume that N ⊂ R q . Then (1.2) acquires the form ∆u = II(du, du) + Z(du(e 1 ) ∧ du(e 2 )) + ∇V (u), where u : M → R q and II denotes the second fundamental form of N in R q . For the equivalence of (1.2) and (2.1) see [2,Lemma 3.8].
In particular, we want to address the question how the regularity of the scalar potential V (u) influences the regularity of the solution of (1.2). To this end, we will make the following definition: Definition 2.1. We call u ∈ W 1,2 (M, N ) a weak solution if it solves (2.1) in a distributional sense.
A similar study has already been performed in [5,6] for harmonic maps with potential, that is critical points of (1.1) with B = 0. Fortunately, by now there exist powerful tools that are well-adapted to (2.1). In the following we will make use of the following regularity result from [12].
where Ω ∈ L 2 (D, so(q) ⊗ R 2 ) and p ∈ (1, 2). Then φ ∈ W 2,p loc (D). In particular, if f = 0, then φ ∈ W 2,p loc for all p ∈ [1, 2) and φ ∈ W 1,q loc for all q ∈ [1, ∞). Moreover, for U ⊂ D, there exist η 0 = η 0 (p, q) > 0 and C = C(p, m, U ) < ∞ such that if Ω L 2 (D) ≤ η 0 , then the following estimate holds In order to be able to apply Theorem 2.2 we need to rewrite the right hand side of (2.1). We denote coordinates in the ambient space R q by (y 1 , y 2 , . . . , y q ). Let ν l , l = n + 1, . . . , q be an orthonormal frame field for the normal bundle T ⊥ N . For X, Y ∈ T y N and ∇ Y ν k = Y i ∂ν k ∂y i we express the second fundamental form as Let D be a domain in M and consider a weak solution of (2.1). We choose local isothermal coordinates z = x + iy, set e 1 = ∂ x , e 2 = ∂ y and use the notation u α = du(e α ). Moreover, note that u α ∈ T N and ν l ∈ T ⊥ N , which implies that where we used (2.4) in the second term on the right hand side. In addition, we note that Z m (du(e 1 ) ∧ du(e 2 )) = Z m (∂ y i ∧ ∂ y j )u i x u j y , m = 1, . . . , q.
By the definition of Z and exploiting the skew-symmetry of the three-form Ω, we find (see also [1]) We are now in the position to show that solutions of (2.1) have a structure such that Theorem 2.2 can be applied. holds.
Proof. By assumption N ⊂ R q is compact, we denote its unit normal field by ν l , l = n + 1, . . . , q. Using (2.5) and (2.6), we denote The skew-symmetry of A m i can be read of from its definition and the properties of Z, see (2.6). By assumption u is a weak solution of (2.1), hence A m i ∈ L 2 (D, so(q) ⊗ R 2 ) completing the proof.
First, we will assume that the scalar potential V (φ) may have as little regularity as possible.
Corollary 2.4. Let (M, h) be a closed Riemannian surface and let N be a compact Riemannian manifold. Assume that u : D → N is a weak solution of (2.1). Fix p ∈ (1, 2) and assume that the scalar potential is of class V ∈ W 1,p (N, R). Then u ∈ W 2,p (M, N ) and u ∈ W Proof. This follows from Theorem 2.2 applied to (2.7) and the Sobolev embedding theorem in dimension two.
One cannot expect to gain more regularity until one assumes that the potential V (u) has a better analytical structure. In the case of a smooth potential V (u) we directly obtain Theorem 1.1.
Proof of Theorem 1.1. This follows from elliptic regularity and a standard bootstrap argument.
We conclude this section with the following "gap-type" theorem.
Proposition 2.5. Let φ be a smooth solution of (2.1) with small energy dφ L 2 < ε. Then the following inequality holds .
The claim follows by applying the Sobolev embedding theorem and choosing ε sufficiently small.
This allows us to draw the following Corollary 2.6. If dφ L 2 is sufficiently small and ∇V for δ sufficiently small then φ must be trivial.
Note that we do not have to make any assumption on V (φ) but only on its gradient.

The heat flow for the full bosonic string
In this section we study the heat flow associated to (1.2) and prove Theorem 1.2. First, we will rewrite the action functional (1.1) in order to obtain a functional that is easier to handle from an analytical point of view. By assumption the manifold N is compact, hence the potential V (φ) satisfies and consider the transformed energy functional The critical points ofS bos (φ, h) and S bos (φ, h) coincide since both action functionals only differ by a constant. This fact is well known in physics: The Lagrangian/Hamiltonian of a mechanical system can be changed by adding a constant since it does not contribute to the equations of motion.
In order to deal with the analytic aspects of (1.3) we again isometrically embedded the target manifold N into R q . Then the corresponding heat-flow acquires the form We will use a subscript t to denote the t-dependence of u. For a derivation of (3.2) see [2,Lemma 4.1]. The existence of a shorttime solution can be obtained by standard methods.
3.1. Energy Estimates. In this subsection we will derive the necessary energy estimates for the study of (3.2). Let us introduce the following notation Here, B R (x) denotes the geodesic ball of radius R around the point x and by ι M we will denote the injectivity radius of M . Note that both of these energies are conformally invariant.
In addition, we introduce the following function space with Q = M × [0, T ) and dQ h = dvol h dt: Due to the variational structure of our problem we have the following Lemma 3.2. Let u t ∈ W be a solution of (3.2). Then the following equality holds Proof. We calculate d dt where we used that II ⊥ ∂ut ∂t . The claim follows by integration with respect to t. The next Lemma is the analogue of Lemma 3.6 from [13].
where the constant C only depends on M .
Proof. We choose a smooth cut-off function η with the following properties where again B R (x 0 ) denotes the geodesic ball of radius R around x 0 ∈ M and C a positive constant. In addition, we choose an orthonormal basis {e α , α = 1, 2} on M such that ∇ eα e β = ∇ ∂t e α = 0 at the considered point. By a direct calculation we obtain ∂ ∂t Multiplying by the cut-off function η 2 and using the evolution equation (3.2) we find d dt Using integration by parts we derive Applying Young's inequality and by the properties of the cut-off function η, we find Integration with respect to t yields the result.
Proposition 3.4. Let u t ∈ W be a solution of (3.2). Moreover, suppose that |B| L ∞ < 1 2 . Then the following monotonicity formulas hold Here, S bos (u 0 , B 2R ) denotes the action functional at time 0 restricted to the ball B 2R .
Proof. This follows from combining (3.3) and (3.4) and making use of the assumptions.
In the following we want to control the energy of u t locally.
Let Ω ∈ R 2 be a bounded domain. Then Ladyzhenskaya's inequality holds, that is Lemma 3.6. Assume that v ∈ W 1,2 (Ω). Then the following inequality holds: In the following we need a local version of Ladyzhenskaya's inequality from above.
Lemma 3.7. Assume that v ∈ W . Then there exists a constant C such that for any R ∈ (0, i M ) the following inequality holds: Proof. A proof can for example be found in [15,Lemma 6.7].
Making use of the Ricci identity we obtain the following formula for v : Here, Scal denotes the scalar curvature of M . As a next step we will control the L 2 -norm of the second derivatives of u t .
Lemma 3.8. Let u t ∈ W be a solution of (3.2) and suppose that (3.7) holds with δ 1 sufficiently small. Moreover, suppose that |B| L ∞ < 1 2 . Then the following inequality holds where the constant C depends on M, N, Proof. By a direct calculation we find d dt By assumption N is compact and we can estimate the Hessian of V by its maximum. Making use of (3.8) and (3.9) we obtain d dt Choosing δ 1 small enough we get the following inequality d dt The claim follows by integration with respect to t.
Using the bound on the second derivatives, we can apply the Sobolev embedding theorem to bound Q |du t | 4 dQ h .
Corollary 3.9. Let u t ∈ W be a solution of (3.2) with δ 1 sufficiently small. Moreover, suppose that |B| L ∞ < 1 2 . Then we have for all t ∈ [0, T 1 ) Proof. The bound follow from (3.8) and the previous estimate.
As a next step we control the L 2 -norm of the derivatives of u t with respect to t.
where the positive constant C depends on M, N, Proof. By a direct calculation using (3.2) we find d dt Again, we can estimate the Hessian of the potential V (u t ) by its maximum since N is compact. Consequently, we obtain d dt To control the second term on the right hand side we use another type of Sobolev inequality (similar to (3.8) for |t − s| ≤ 1), that is Using (3.11) and integrating over a small time interval t − s < z, we can absorb part of the right hand side in the left and obtain Finally, we estimate the infimum by the mean value, more precisely sup Hence, we get the desired bound.
Lemma 3.11. Let u t ∈ W be a solution of (3.2) with |B| L ∞ < 1 2 . As long as δ 1 is sufficiently small we have the following bound (3.13) The constant C depends on M, N, Proof. Using (3.2) and (3.9) we obtain the following inequality Since we are assuming the potential V (u) to be smooth and N to be compact we can easily estimate |∇V (u)| 2 . Applying (3.8) and (3.12) with δ 1 sufficiently small yields the claim.
Proposition 3.12 (Higher Regularity). Let u t ∈ W be a solution of (3.2). As long as δ 1 is small enough the solution u t of (3.2) is smooth.
Proof. This follows from standard regularity theory arguments, see for example [15,Lemma 6.11] and references therein for more details.
Let us close this section with the following remarks.
(1) The fact that u t is smooth as long as δ 1 is sufficiently small relies on the fact that we have a smooth scalar potential V (u). If we would assume lower regularity of V (u) then u t would also have less regularity. We can use the parabolicity of (3.2) to smoothen out distributional initial data, but the parabolicity cannot compensate for a potential of bad regularity.
In order to achieve that u ∈ W 2,2 (M, N ) we have to require that V ∈ C 2 (N, R). By the Sobolev embedding theorem we then get that u is continuous, to gain more regularity we need better regularity of V (u).
(2) In the case of a smooth heat flow one can also consider the case of a non-compact target N and use the potential V (u) to constrain the image of M under u t to a compact set. However, this argument makes use of the maximum principle, which we cannot apply in our case.

Longtime Existence.
In this section we establish the existence of a unique global weak solution to (3.2) for all times t ∈ [0, ∞). Moreover, we will show that only finitely many singularities will occur along the flow. First, we will give a uniqueness result.
Proposition 3.14. Let u t , v t ∈ W be two solutions of (3.2) and suppose that |B| L ∞ < 1 2 . If their initial data coincides, that is u 0 = v 0 , then u t = v t for all t ∈ [0, T ).
Proof. Throughout the proof C will denote a universal constant that may change from line to line. Let u t , v t be two solutions of (3.2). We set w t := u t − v t . By projecting to a tubular neighborhood II(u t )(du t , du t ), Z(u t )(du t (e 1 ) ∧ du t (e 2 )) and ∇V (u t ) can be thought of as vectorvalued functions in R q , for more details see [2,Lemma 4.8]. Exploiting this fact a direct computation yields Rewriting and similarly for the terms containing Z we find d dt This leads to the following inequality By assumption the scalar potential V (u) is smooth such that we can apply the mean-value theorem and estimate Taking the limit T → 0 and applying the Gronwall inequality allows us to conclude the claim.
Remark 3.15. The proof of the previous Proposition requires the potential V (φ) to be sufficiently regular such that we can apply the mean-value theorem. If we would allow for a potential with less regularity it does not seem to be possible to prove uniqueness of the solution of (3.2).
By the same strategy as in the case of standard harmonic maps we can establish the long-time existence of (3.2).
Proposition 3.17. Let u t ∈ W be a solution of (3.2). Suppose that |B| L ∞ < 1 2 and that the scalar potential V (u) is sufficiently small, that is (3.14) Then there are only finitely many singular points ( Proof. We follow the presentation in [10, p.138] for the harmonic map heat flow. We assume that T 0 > 0 is the first singular time and define the singular set as Now, let {x j } K j=1 be any finite subset of Z(u, T 0 ). Then we have for R > 0 lim sup We choose R > 0 such that all the B 2R (x j ), 1 ≤ j ≤ K are mutually disjoint. Then, we have by (3.6) which implies the finiteness of the singular set Z(u, T 0 ). Next, we show that there are only finitely many singular spatial points. We set and calculate Suppose T 0 < . . . < T j are j singular times and by K 0 , . . . , K j we denote the number of singular points at each singular time. Set By iterating (3.15) we get which can be rearranged as In order to conclude that the number of singularities is finite we have to ensure that where K i ≥ 1. Making use of the assumptions we obtain the claim.
(1) A careful analysis of the last proof reveals that the condition on the smallness of the scalar potential (3.14) is actually only needed at the singular times T i , that is However, it seems rather unlikely that this condition can be satisfied in general. (2) In the case of the standard harmonic map heat flow we have δ 3 = 1 and V (u t ) = 0 such that the bound on the number of singularities reduces to Moreover, in contrast to the harmonic map heat flow the number of singularities also depends on the metric on M .
(3) We want to emphasize that we require the potential V (u) itself to be sufficiently small and do not demand any smallness of its gradient. This is what one expects from a mathematics perspective since the potential itself enters the action functional (1.1). However, from a physics perspective the important quantity is the gradient of the potential since it corresponds to the force acting on a system.
There is a second way of controlling the number of singularities. Instead of requiring the potential V (u t ) to be sufficiently small as in (3.14) we can exploit the fact that (1.1) is not conformally invariant. More precisely, if we perform a rescaling of the metrich = ah, where a is supposed to be a positive real number, then the first two-terms of (1.1) are not affected, whereas the scalar potential gets rescaled. More precisely, we find In terms of the rescaled metrich = ah the smallness condition (3.14) can be expressed as Choosing a 2 large enough we can achieve to have a finite number of singularities without posing any smallness condition on the scalar potential V (φ). However, the finiteness of the number of singularities now depends on the rescaled metric on the domain M .

3.3.
Convergence. In this subsection we address the issue of convergence of (3.2).
Proposition 3.20. Let u t ∈ W be a solution of (3.2) and suppose that |B| L ∞ < 1 2 . Then there exists a sequence t k such that u t k converges weakly in W 1,2 (M, N ) and strongly in the space W 2,2 loc (M \ {x k , t k = ∞}) to a solution of (2.1). The limiting map u ∞ is smooth on M \ {x 1 , . . . , x k }.
Proof. First, we suppose that T = ∞ is non-singular, that is lim sup t→∞ ( sup x∈M E(u t , B R (x))) < δ 1 for some R > 0. Since we have a uniform bound on the L 2 norm of the t derivative of u t by Lemma 3.3, we can achieve for t k → ∞ suitably that By (3.13) we have a bound on the second derivatives Moreover, by the Rellich-Kondrachov embedding theorem we have for any p < ∞. We get convergence of the evolution equation (3.2) in L 2 , consequently u ∞ is a solution of (2.1) satisfying u ∞ ∈ W 2,2 (M, N ). If T = ∞ is singular, that is at the points {x 1 , . . . , for all R > 0, then for suitable numbers t k → ∞ the family u t k will be bounded in W 2,2 loc (M, N ) on the set M \ {x 1 , . . . , x k }. Consequently, the family u t k will accumulate as follows We setM := M \ {x 1 , . . . , x k }. Since we have enough control over the energy of u ∞ by (3.5), that is E(u ∞ ) ≤ C, we can apply Theorem 1.1 finishing the proof.
This completes the proof of Theorem 1.2.

Blowup analysis.
In order to discuss a blowup analysis of the singular points recall the definition of the parabolic cylinder For simplicity, assume that (0, 0) is a singular point of u ∈ C ∞ (P 1 (0, 0) \ {0, 0}, N ). Then there exist r k → 0 as k → ∞ and z k = (x k , t k ) with x k → 0, t k → 0 as k → ∞. It is easy to check that v k satisfies ∂v k ∂t = ∆v k − II(dv k , dv k ) − Z(dv k (e 1 ) ∧ dv k (e 2 )) − 1 r 2 k ∇V (v k ).
(3. 16) In the limit k → ∞, we would have P r −1 k → R 2 × R − , but it is obvious that (3.16) blows up as k → ∞. This behavior should be expected since the scalar potential V (u) breaks the conformal invariance of the energy functional (1.1). However, if V (u) = 0 the energy functional (1.1) is invariant under conformal transformations on the domain and we find that E(u t k , B r k (x k )) = sup z=(x,t)∈P 1 ,−1≤t≤t k E(u t , B r k (x)) = δ 1 C for C > 0 sufficiently large. Assume that t k − 4r 2 k ≥ −1. Moreover, we have and also E(v k (t)) ≤ E(u 0 ), −r −2 k ≤ t ≤ 0, sup (x,t)∈P k E(v k (t), B 2 (x)) ≤ C sup (x,t)∈P 1 E(u t , B r k (x)) ≤ δ 1 .
Consequently, we can take the limit k → ∞ and v k converges to some limiting map ω. Then, ω ∈ C ∞ (R 2 × (−∞, 0), N ) solves 0 = ∆ω − II(dω, dω) − Z(dω(e 1 ) ∧ dω(e 2 )) since ∂ t ω = 0. Using the conformal invariance we perform a stereographic projection to S 2 and obtain a solution of τ (φ) = Z(dφ(e 1 ) ∧ dφ(e 2 )), where φ : S 2 → N . Making use of a Theorem of Grüter [9] we can remove the singular points that we get from the stereographic projection. Hence, we get a variant of the usual bubbling that is well known in the standard harmonic map heat flow. is known as prescribed curvature equation. Thus, the bubbling described above in the case of V (φ) = 0 yields maps with prescribed mean curvature from S 2 . However, the condition |B| L ∞ < 1 2 that we needed to impose does not seem to have a natural geometric interpretation.

3.5.
Qualitative properties of the limiting map. Let us briefly discuss the qualitative behavior of solutions to (1.2).