Citation Context

...4 Elsevier B.V. All rights reserved. 3536 H. Guo et al. / Stochastic Processes and their Applications 124 (2014) 3535–3552 and this has been done in a number of papers, e.g. [2,8,14,15,29,33]. In particular, stochastic representation formulae for the differential of harmonic maps have turned out to be a powerful tool to prove Liouville theorems, i.e. theorems stating that harmonic maps in a certain class of maps and under certain topological or geometric constraints are necessarily constant [33]. Due to Perelman’s proof of the geometrization and hence the Poincare conjecture using Ricci flow [25,27,26], there is now a strong interest in studying manifolds M with time-dependent geometry. In such a context, the notion of harmonic map turns out to be no longer appropriate; however, it is natural to study space–time harmonic maps which by time reversal provide solutions to the harmonic map heat flow (or nonlinear heat equation), see e.g. [18,23,35]. The behavior of (positive) solutions to the linear heat equation under Ricci flow has been intensively studied during the last decade, e.g. [5,21,22]. It is clear from the static case that in the nonlinear situation under Ricci flow also the geometr...

Citation Context

...rems under appropriate curvature conditions. Space–time harmonic mappings which are defined globally in time correspond to ancient solutions to the harmonic map heat flow. As corollaries, we establish triviality of such ancient solutions in a variety of different situations. c 2014 Elsevier B.V. All rights reserved. MSC: 53C44; 58J65 Keywords: Harmonic map heat flow; Stochastic analysis on manifolds; Time-dependent geometry 1. Introduction A smooth mapping u : M → N between Riemannian manifolds (M, g) and (N , h) is said to be harmonic if its tension field ∆g,hu ≡ trace∇du vanishes, see e.g. [9,19]. Since harmonic maps are characterized by the property that they map M-valued Brownian motions to N -valued martingales (see e.g. [11, Satz 7.157(ii)]), it is natural to study them using stochastic methods, ∗ Corresponding author. Tel.: +352 4666445867. E-mail addresses: guo@wzu.edu.cn (H. Guo), robert.philipowski@uni.lu (R. Philipowski), anton.thalmaier@uni.lu (A. Thalmaier). http://dx.doi.org/10.1016/j.spa.2014.06.004 0304-4149/ c 2014 Elsevier B.V. All rights reserved. 3536 H. Guo et al. / Stochastic Processes and their Applications 124 (2014) 3535–3552 and this has been done in a number ...

Citation Context

...4 Elsevier B.V. All rights reserved. 3536 H. Guo et al. / Stochastic Processes and their Applications 124 (2014) 3535–3552 and this has been done in a number of papers, e.g. [2,8,14,15,29,33]. In particular, stochastic representation formulae for the differential of harmonic maps have turned out to be a powerful tool to prove Liouville theorems, i.e. theorems stating that harmonic maps in a certain class of maps and under certain topological or geometric constraints are necessarily constant [33]. Due to Perelman’s proof of the geometrization and hence the Poincare conjecture using Ricci flow [25,27,26], there is now a strong interest in studying manifolds M with time-dependent geometry. In such a context, the notion of harmonic map turns out to be no longer appropriate; however, it is natural to study space–time harmonic maps which by time reversal provide solutions to the harmonic map heat flow (or nonlinear heat equation), see e.g. [18,23,35]. The behavior of (positive) solutions to the linear heat equation under Ricci flow has been intensively studied during the last decade, e.g. [5,21,22]. It is clear from the static case that in the nonlinear situation under Ricci flow also the geometr...

Citation Context

...4 Elsevier B.V. All rights reserved. 3536 H. Guo et al. / Stochastic Processes and their Applications 124 (2014) 3535–3552 and this has been done in a number of papers, e.g. [2,8,14,15,29,33]. In particular, stochastic representation formulae for the differential of harmonic maps have turned out to be a powerful tool to prove Liouville theorems, i.e. theorems stating that harmonic maps in a certain class of maps and under certain topological or geometric constraints are necessarily constant [33]. Due to Perelman’s proof of the geometrization and hence the Poincare conjecture using Ricci flow [25,27,26], there is now a strong interest in studying manifolds M with time-dependent geometry. In such a context, the notion of harmonic map turns out to be no longer appropriate; however, it is natural to study space–time harmonic maps which by time reversal provide solutions to the harmonic map heat flow (or nonlinear heat equation), see e.g. [18,23,35]. The behavior of (positive) solutions to the linear heat equation under Ricci flow has been intensively studied during the last decade, e.g. [5,21,22]. It is clear from the static case that in the nonlinear situation under Ricci flow also the geometr...

Citation Context

... Due to Perelman’s proof of the geometrization and hence the Poincare conjecture using Ricci flow [25,27,26], there is now a strong interest in studying manifolds M with time-dependent geometry. In such a context, the notion of harmonic map turns out to be no longer appropriate; however, it is natural to study space–time harmonic maps which by time reversal provide solutions to the harmonic map heat flow (or nonlinear heat equation), see e.g. [18,23,35]. The behavior of (positive) solutions to the linear heat equation under Ricci flow has been intensively studied during the last decade, e.g. [5,21,22]. It is clear from the static case that in the nonlinear situation under Ricci flow also the geometry of the target space will naturally play a crucial role, see e.g. [19,31]. Building on our previous work on martingales on manifolds with time-dependent connection [10], we establish stochastic representation formulae for space–time harmonic maps and solutions to the harmonic map heat flow defined on a manifold with time-dependent metric. We then apply these formulae to prove Liouville theorems for space–time harmonic maps and ancient solutions to the harmonic map heat flow under appropriate cu...

Citation Context

...dependent geometry. In such a context, the notion of harmonic map turns out to be no longer appropriate; however, it is natural to study space–time harmonic maps which by time reversal provide solutions to the harmonic map heat flow (or nonlinear heat equation), see e.g. [18,23,35]. The behavior of (positive) solutions to the linear heat equation under Ricci flow has been intensively studied during the last decade, e.g. [5,21,22]. It is clear from the static case that in the nonlinear situation under Ricci flow also the geometry of the target space will naturally play a crucial role, see e.g. [19,31]. Building on our previous work on martingales on manifolds with time-dependent connection [10], we establish stochastic representation formulae for space–time harmonic maps and solutions to the harmonic map heat flow defined on a manifold with time-dependent metric. We then apply these formulae to prove Liouville theorems for space–time harmonic maps and ancient solutions to the harmonic map heat flow under appropriate curvature conditions. 2. Stochastic representation formulae Let M be a differentiable manifold equipped with a smooth family g(t) of Riemannian metrics (t ∈ (T0, T ] with T0 < ...

Citation Context

...0 + k(r0) 2r0 , which completes the proof. Remark 3.3. The key ingredient of the proof above is estimate (3.12) for the radial drift of Brownian motion, which should be seen as a parabolic version of the Laplacian comparison theorem for evolving manifolds. In the case of a fixed metric with non-negative Ricci curvature the Laplacian comparison theorem however provides the much better estimate 1 2 ∆ρ ≤ d − 1 2ρ . (3.13) Since in many respects manifolds evolving under backward super Ricci flow behave in a similar way as manifolds with a fixed metric of non-negative Ricci curvature (see e.g. [20] or [34, Section 6.5]), one might expect that an estimate similar to (3.13) also holds under backward super Ricci flow. This, however, is not the case, as the following example shows. Example 3.4 (Brownian Motion on Hamilton’s Cigar). Let M = R2 be equipped with the timedependent metric g(t, x) := 1 e−2t + |x |2 g eucl(x), where g eucl denotes the standard metric on R2. As shown in [6, Section 4.3], the family (g(t))t∈R is an eternal solution of the backward Ricci flow, called “Hamilton’s cigar” or “Witten’s black hole”. By elementary calculations one obtains ρ(t, x) = arcsinh(et |x |), and co...

Citation Context

... holds for all (t, z) ∈ (−∞, T ] × M . Corollary 4.2. Suppose that M is connected and that ∂g ∂t ≥ −Ricg(t) on (−∞, T ] × M (forward super Ricci flow). Assume that for each x ∈ M there exists r0 > 0 such that sup |Ric(t, y) |: t ∈ (−∞, T ], dg(t)(x, y) ≤ r0 (the analogue of the constant C(x, r0) defined in (3.9)) is finite, and that N is simply-connected and has non-positive sectional curvatures. Then any ancient solution of sub-square-root growth u : (−∞, T ] × M → N to the harmonic map heat flow is constant. Remark 4.3. Theorem 4.1 should be compared with S.-Y. Cheng’s Liouville theorem [4] which gives an analogous statement for harmonic maps of sublinear growth when M is equipped with a fixed metric of non-negative Ricci curvature, see [33, Corollary 5.10] and [30] for stochastic proofs. All these proofs depend crucially on the Laplacian comparison theorem. In the case of backward super Ricci flow the stronger assumption of sub-square-root growth is needed because estimate (3.10) is weaker than estimate (3.11) which holds in the case of a fixed metric with non-negative Ricci curvature, see the discussion in Remark 3.2. H. Guo et al. / Stochastic Processes and their Applications...

Citation Context

...rems under appropriate curvature conditions. Space–time harmonic mappings which are defined globally in time correspond to ancient solutions to the harmonic map heat flow. As corollaries, we establish triviality of such ancient solutions in a variety of different situations. c 2014 Elsevier B.V. All rights reserved. MSC: 53C44; 58J65 Keywords: Harmonic map heat flow; Stochastic analysis on manifolds; Time-dependent geometry 1. Introduction A smooth mapping u : M → N between Riemannian manifolds (M, g) and (N , h) is said to be harmonic if its tension field ∆g,hu ≡ trace∇du vanishes, see e.g. [9,19]. Since harmonic maps are characterized by the property that they map M-valued Brownian motions to N -valued martingales (see e.g. [11, Satz 7.157(ii)]), it is natural to study them using stochastic methods, ∗ Corresponding author. Tel.: +352 4666445867. E-mail addresses: guo@wzu.edu.cn (H. Guo), robert.philipowski@uni.lu (R. Philipowski), anton.thalmaier@uni.lu (A. Thalmaier). http://dx.doi.org/10.1016/j.spa.2014.06.004 0304-4149/ c 2014 Elsevier B.V. All rights reserved. 3536 H. Guo et al. / Stochastic Processes and their Applications 124 (2014) 3535–3552 and this has been done in a number ...

Citation Context

...ned on R+×M , and vice versa. For this reason, in the sequel, we formulate our H. Guo et al. / Stochastic Processes and their Applications 124 (2014) 3535–3552 3537 results for space–time harmonic maps; however the statements immediately apply to solutions of the harmonic map heat flow by time reversal. From now on, let (g(t))t≥0 be a smooth family of Riemannian metrics on M and (N , h) be a Riemannian manifold. Let u : [0, ∞) × M → N be a space–time harmonic map in the sense that ∂u ∂t + 1 2 ∆g(t),hu = 0. (2.2) Notation 2.2. Fixing a point x ∈ M , let (X t )t≥0 be a g(t)-Brownian motion on M [1,7,17, 16,24] starting at x , and consider the image process X t := u(t, X t ) taking values in the target manifold N . As in [10, Theorem 9.3 and Remark 9.4] let Θ0,t : Tx M → TX t M be the damped parallel transport along X , defined by the covariant equation d //Riem0,t −1 Θ0,t = − 1 2 //Riem0,t −1 − ∂g ∂t + Ricg(t) # Θ0,t dt, Θ0,0 = idTx M , where //Riem0,t : Tx M → TX t M is the Riemann-parallel transport along X , see [10, Definition 3.3]. Similarly, in terms of the Riemann curvature tensor R on N , let Θ0,t : TX0 N → TX t N be the damped parallel transport along X , defined by the c...

Citation Context

...ned on R+×M , and vice versa. For this reason, in the sequel, we formulate our H. Guo et al. / Stochastic Processes and their Applications 124 (2014) 3535–3552 3537 results for space–time harmonic maps; however the statements immediately apply to solutions of the harmonic map heat flow by time reversal. From now on, let (g(t))t≥0 be a smooth family of Riemannian metrics on M and (N , h) be a Riemannian manifold. Let u : [0, ∞) × M → N be a space–time harmonic map in the sense that ∂u ∂t + 1 2 ∆g(t),hu = 0. (2.2) Notation 2.2. Fixing a point x ∈ M , let (X t )t≥0 be a g(t)-Brownian motion on M [1,7,17, 16,24] starting at x , and consider the image process X t := u(t, X t ) taking values in the target manifold N . As in [10, Theorem 9.3 and Remark 9.4] let Θ0,t : Tx M → TX t M be the damped parallel transport along X , defined by the covariant equation d //Riem0,t −1 Θ0,t = − 1 2 //Riem0,t −1 − ∂g ∂t + Ricg(t) # Θ0,t dt, Θ0,0 = idTx M , where //Riem0,t : Tx M → TX t M is the Riemann-parallel transport along X , see [10, Definition 3.3]. Similarly, in terms of the Riemann curvature tensor R on N , let Θ0,t : TX0 N → TX t N be the damped parallel transport along X , defined by the c...

Citation Context

... Due to Perelman’s proof of the geometrization and hence the Poincare conjecture using Ricci flow [25,27,26], there is now a strong interest in studying manifolds M with time-dependent geometry. In such a context, the notion of harmonic map turns out to be no longer appropriate; however, it is natural to study space–time harmonic maps which by time reversal provide solutions to the harmonic map heat flow (or nonlinear heat equation), see e.g. [18,23,35]. The behavior of (positive) solutions to the linear heat equation under Ricci flow has been intensively studied during the last decade, e.g. [5,21,22]. It is clear from the static case that in the nonlinear situation under Ricci flow also the geometry of the target space will naturally play a crucial role, see e.g. [19,31]. Building on our previous work on martingales on manifolds with time-dependent connection [10], we establish stochastic representation formulae for space–time harmonic maps and solutions to the harmonic map heat flow defined on a manifold with time-dependent metric. We then apply these formulae to prove Liouville theorems for space–time harmonic maps and ancient solutions to the harmonic map heat flow under appropriate cu...

Citation Context

... maps are characterized by the property that they map M-valued Brownian motions to N -valued martingales (see e.g. [11, Satz 7.157(ii)]), it is natural to study them using stochastic methods, ∗ Corresponding author. Tel.: +352 4666445867. E-mail addresses: guo@wzu.edu.cn (H. Guo), robert.philipowski@uni.lu (R. Philipowski), anton.thalmaier@uni.lu (A. Thalmaier). http://dx.doi.org/10.1016/j.spa.2014.06.004 0304-4149/ c 2014 Elsevier B.V. All rights reserved. 3536 H. Guo et al. / Stochastic Processes and their Applications 124 (2014) 3535–3552 and this has been done in a number of papers, e.g. [2,8,14,15,29,33]. In particular, stochastic representation formulae for the differential of harmonic maps have turned out to be a powerful tool to prove Liouville theorems, i.e. theorems stating that harmonic maps in a certain class of maps and under certain topological or geometric constraints are necessarily constant [33]. Due to Perelman’s proof of the geometrization and hence the Poincare conjecture using Ricci flow [25,27,26], there is now a strong interest in studying manifolds M with time-dependent geometry. In such a context, the notion of harmonic map turns out to be no longer appropriate; however, ...

Citation Context

... Due to Perelman’s proof of the geometrization and hence the Poincare conjecture using Ricci flow [25,27,26], there is now a strong interest in studying manifolds M with time-dependent geometry. In such a context, the notion of harmonic map turns out to be no longer appropriate; however, it is natural to study space–time harmonic maps which by time reversal provide solutions to the harmonic map heat flow (or nonlinear heat equation), see e.g. [18,23,35]. The behavior of (positive) solutions to the linear heat equation under Ricci flow has been intensively studied during the last decade, e.g. [5,21,22]. It is clear from the static case that in the nonlinear situation under Ricci flow also the geometry of the target space will naturally play a crucial role, see e.g. [19,31]. Building on our previous work on martingales on manifolds with time-dependent connection [10], we establish stochastic representation formulae for space–time harmonic maps and solutions to the harmonic map heat flow defined on a manifold with time-dependent metric. We then apply these formulae to prove Liouville theorems for space–time harmonic maps and ancient solutions to the harmonic map heat flow under appropriate cu...

Citation Context

...ned on R+×M , and vice versa. For this reason, in the sequel, we formulate our H. Guo et al. / Stochastic Processes and their Applications 124 (2014) 3535–3552 3537 results for space–time harmonic maps; however the statements immediately apply to solutions of the harmonic map heat flow by time reversal. From now on, let (g(t))t≥0 be a smooth family of Riemannian metrics on M and (N , h) be a Riemannian manifold. Let u : [0, ∞) × M → N be a space–time harmonic map in the sense that ∂u ∂t + 1 2 ∆g(t),hu = 0. (2.2) Notation 2.2. Fixing a point x ∈ M , let (X t )t≥0 be a g(t)-Brownian motion on M [1,7,17, 16,24] starting at x , and consider the image process X t := u(t, X t ) taking values in the target manifold N . As in [10, Theorem 9.3 and Remark 9.4] let Θ0,t : Tx M → TX t M be the damped parallel transport along X , defined by the covariant equation d //Riem0,t −1 Θ0,t = − 1 2 //Riem0,t −1 − ∂g ∂t + Ricg(t) # Θ0,t dt, Θ0,0 = idTx M , where //Riem0,t : Tx M → TX t M is the Riemann-parallel transport along X , see [10, Definition 3.3]. Similarly, in terms of the Riemann curvature tensor R on N , let Θ0,t : TX0 N → TX t N be the damped parallel transport along X , defined by the c...

Citation Context

... maps are characterized by the property that they map M-valued Brownian motions to N -valued martingales (see e.g. [11, Satz 7.157(ii)]), it is natural to study them using stochastic methods, ∗ Corresponding author. Tel.: +352 4666445867. E-mail addresses: guo@wzu.edu.cn (H. Guo), robert.philipowski@uni.lu (R. Philipowski), anton.thalmaier@uni.lu (A. Thalmaier). http://dx.doi.org/10.1016/j.spa.2014.06.004 0304-4149/ c 2014 Elsevier B.V. All rights reserved. 3536 H. Guo et al. / Stochastic Processes and their Applications 124 (2014) 3535–3552 and this has been done in a number of papers, e.g. [2,8,14,15,29,33]. In particular, stochastic representation formulae for the differential of harmonic maps have turned out to be a powerful tool to prove Liouville theorems, i.e. theorems stating that harmonic maps in a certain class of maps and under certain topological or geometric constraints are necessarily constant [33]. Due to Perelman’s proof of the geometrization and hence the Poincare conjecture using Ricci flow [25,27,26], there is now a strong interest in studying manifolds M with time-dependent geometry. In such a context, the notion of harmonic map turns out to be no longer appropriate; however, ...

Citation Context

... maps are characterized by the property that they map M-valued Brownian motions to N -valued martingales (see e.g. [11, Satz 7.157(ii)]), it is natural to study them using stochastic methods, ∗ Corresponding author. Tel.: +352 4666445867. E-mail addresses: guo@wzu.edu.cn (H. Guo), robert.philipowski@uni.lu (R. Philipowski), anton.thalmaier@uni.lu (A. Thalmaier). http://dx.doi.org/10.1016/j.spa.2014.06.004 0304-4149/ c 2014 Elsevier B.V. All rights reserved. 3536 H. Guo et al. / Stochastic Processes and their Applications 124 (2014) 3535–3552 and this has been done in a number of papers, e.g. [2,8,14,15,29,33]. In particular, stochastic representation formulae for the differential of harmonic maps have turned out to be a powerful tool to prove Liouville theorems, i.e. theorems stating that harmonic maps in a certain class of maps and under certain topological or geometric constraints are necessarily constant [33]. Due to Perelman’s proof of the geometrization and hence the Poincare conjecture using Ricci flow [25,27,26], there is now a strong interest in studying manifolds M with time-dependent geometry. In such a context, the notion of harmonic map turns out to be no longer appropriate; however, ...

Citation Context

....e. theorems stating that harmonic maps in a certain class of maps and under certain topological or geometric constraints are necessarily constant [33]. Due to Perelman’s proof of the geometrization and hence the Poincare conjecture using Ricci flow [25,27,26], there is now a strong interest in studying manifolds M with time-dependent geometry. In such a context, the notion of harmonic map turns out to be no longer appropriate; however, it is natural to study space–time harmonic maps which by time reversal provide solutions to the harmonic map heat flow (or nonlinear heat equation), see e.g. [18,23,35]. The behavior of (positive) solutions to the linear heat equation under Ricci flow has been intensively studied during the last decade, e.g. [5,21,22]. It is clear from the static case that in the nonlinear situation under Ricci flow also the geometry of the target space will naturally play a crucial role, see e.g. [19,31]. Building on our previous work on martingales on manifolds with time-dependent connection [10], we establish stochastic representation formulae for space–time harmonic maps and solutions to the harmonic map heat flow defined on a manifold with time-dependent metric. We then...

Citation Context

....e. theorems stating that harmonic maps in a certain class of maps and under certain topological or geometric constraints are necessarily constant [33]. Due to Perelman’s proof of the geometrization and hence the Poincare conjecture using Ricci flow [25,27,26], there is now a strong interest in studying manifolds M with time-dependent geometry. In such a context, the notion of harmonic map turns out to be no longer appropriate; however, it is natural to study space–time harmonic maps which by time reversal provide solutions to the harmonic map heat flow (or nonlinear heat equation), see e.g. [18,23,35]. The behavior of (positive) solutions to the linear heat equation under Ricci flow has been intensively studied during the last decade, e.g. [5,21,22]. It is clear from the static case that in the nonlinear situation under Ricci flow also the geometry of the target space will naturally play a crucial role, see e.g. [19,31]. Building on our previous work on martingales on manifolds with time-dependent connection [10], we establish stochastic representation formulae for space–time harmonic maps and solutions to the harmonic map heat flow defined on a manifold with time-dependent metric. We then...

Citation Context

....e. theorems stating that harmonic maps in a certain class of maps and under certain topological or geometric constraints are necessarily constant [33]. Due to Perelman’s proof of the geometrization and hence the Poincare conjecture using Ricci flow [25,27,26], there is now a strong interest in studying manifolds M with time-dependent geometry. In such a context, the notion of harmonic map turns out to be no longer appropriate; however, it is natural to study space–time harmonic maps which by time reversal provide solutions to the harmonic map heat flow (or nonlinear heat equation), see e.g. [18,23,35]. The behavior of (positive) solutions to the linear heat equation under Ricci flow has been intensively studied during the last decade, e.g. [5,21,22]. It is clear from the static case that in the nonlinear situation under Ricci flow also the geometry of the target space will naturally play a crucial role, see e.g. [19,31]. Building on our previous work on martingales on manifolds with time-dependent connection [10], we establish stochastic representation formulae for space–time harmonic maps and solutions to the harmonic map heat flow defined on a manifold with time-dependent metric. We then...

Citation Context

...riate; however, it is natural to study space–time harmonic maps which by time reversal provide solutions to the harmonic map heat flow (or nonlinear heat equation), see e.g. [18,23,35]. The behavior of (positive) solutions to the linear heat equation under Ricci flow has been intensively studied during the last decade, e.g. [5,21,22]. It is clear from the static case that in the nonlinear situation under Ricci flow also the geometry of the target space will naturally play a crucial role, see e.g. [19,31]. Building on our previous work on martingales on manifolds with time-dependent connection [10], we establish stochastic representation formulae for space–time harmonic maps and solutions to the harmonic map heat flow defined on a manifold with time-dependent metric. We then apply these formulae to prove Liouville theorems for space–time harmonic maps and ancient solutions to the harmonic map heat flow under appropriate curvature conditions. 2. Stochastic representation formulae Let M be a differentiable manifold equipped with a smooth family g(t) of Riemannian metrics (t ∈ (T0, T ] with T0 < T ), and let (N , h) be a Riemannian manifold. Let u : (T0, T ] × M → N be a solution to the ha...

Citation Context

...ned on R+×M , and vice versa. For this reason, in the sequel, we formulate our H. Guo et al. / Stochastic Processes and their Applications 124 (2014) 3535–3552 3537 results for space–time harmonic maps; however the statements immediately apply to solutions of the harmonic map heat flow by time reversal. From now on, let (g(t))t≥0 be a smooth family of Riemannian metrics on M and (N , h) be a Riemannian manifold. Let u : [0, ∞) × M → N be a space–time harmonic map in the sense that ∂u ∂t + 1 2 ∆g(t),hu = 0. (2.2) Notation 2.2. Fixing a point x ∈ M , let (X t )t≥0 be a g(t)-Brownian motion on M [1,7,17, 16,24] starting at x , and consider the image process X t := u(t, X t ) taking values in the target manifold N . As in [10, Theorem 9.3 and Remark 9.4] let Θ0,t : Tx M → TX t M be the damped parallel transport along X , defined by the covariant equation d //Riem0,t −1 Θ0,t = − 1 2 //Riem0,t −1 − ∂g ∂t + Ricg(t) # Θ0,t dt, Θ0,0 = idTx M , where //Riem0,t : Tx M → TX t M is the Riemann-parallel transport along X , see [10, Definition 3.3]. Similarly, in terms of the Riemann curvature tensor R on N , let Θ0,t : TX0 N → TX t N be the damped parallel transport along X , defined by the c...

Citation Context

...ned on R+×M , and vice versa. For this reason, in the sequel, we formulate our H. Guo et al. / Stochastic Processes and their Applications 124 (2014) 3535–3552 3537 results for space–time harmonic maps; however the statements immediately apply to solutions of the harmonic map heat flow by time reversal. From now on, let (g(t))t≥0 be a smooth family of Riemannian metrics on M and (N , h) be a Riemannian manifold. Let u : [0, ∞) × M → N be a space–time harmonic map in the sense that ∂u ∂t + 1 2 ∆g(t),hu = 0. (2.2) Notation 2.2. Fixing a point x ∈ M , let (X t )t≥0 be a g(t)-Brownian motion on M [1,7,17, 16,24] starting at x , and consider the image process X t := u(t, X t ) taking values in the target manifold N . As in [10, Theorem 9.3 and Remark 9.4] let Θ0,t : Tx M → TX t M be the damped parallel transport along X , defined by the covariant equation d //Riem0,t −1 Θ0,t = − 1 2 //Riem0,t −1 − ∂g ∂t + Ricg(t) # Θ0,t dt, Θ0,0 = idTx M , where //Riem0,t : Tx M → TX t M is the Riemann-parallel transport along X , see [10, Definition 3.3]. Similarly, in terms of the Riemann curvature tensor R on N , let Θ0,t : TX0 N → TX t N be the damped parallel transport along X , defined by the c...

Citation Context

... maps are characterized by the property that they map M-valued Brownian motions to N -valued martingales (see e.g. [11, Satz 7.157(ii)]), it is natural to study them using stochastic methods, ∗ Corresponding author. Tel.: +352 4666445867. E-mail addresses: guo@wzu.edu.cn (H. Guo), robert.philipowski@uni.lu (R. Philipowski), anton.thalmaier@uni.lu (A. Thalmaier). http://dx.doi.org/10.1016/j.spa.2014.06.004 0304-4149/ c 2014 Elsevier B.V. All rights reserved. 3536 H. Guo et al. / Stochastic Processes and their Applications 124 (2014) 3535–3552 and this has been done in a number of papers, e.g. [2,8,14,15,29,33]. In particular, stochastic representation formulae for the differential of harmonic maps have turned out to be a powerful tool to prove Liouville theorems, i.e. theorems stating that harmonic maps in a certain class of maps and under certain topological or geometric constraints are necessarily constant [33]. Due to Perelman’s proof of the geometrization and hence the Poincare conjecture using Ricci flow [25,27,26], there is now a strong interest in studying manifolds M with time-dependent geometry. In such a context, the notion of harmonic map turns out to be no longer appropriate; however, ...

Citation Context

... maps are characterized by the property that they map M-valued Brownian motions to N -valued martingales (see e.g. [11, Satz 7.157(ii)]), it is natural to study them using stochastic methods, ∗ Corresponding author. Tel.: +352 4666445867. E-mail addresses: guo@wzu.edu.cn (H. Guo), robert.philipowski@uni.lu (R. Philipowski), anton.thalmaier@uni.lu (A. Thalmaier). http://dx.doi.org/10.1016/j.spa.2014.06.004 0304-4149/ c 2014 Elsevier B.V. All rights reserved. 3536 H. Guo et al. / Stochastic Processes and their Applications 124 (2014) 3535–3552 and this has been done in a number of papers, e.g. [2,8,14,15,29,33]. In particular, stochastic representation formulae for the differential of harmonic maps have turned out to be a powerful tool to prove Liouville theorems, i.e. theorems stating that harmonic maps in a certain class of maps and under certain topological or geometric constraints are necessarily constant [33]. Due to Perelman’s proof of the geometrization and hence the Poincare conjecture using Ricci flow [25,27,26], there is now a strong interest in studying manifolds M with time-dependent geometry. In such a context, the notion of harmonic map turns out to be no longer appropriate; however, ...

Citation Context

... there exists r0 > 0 such that sup |Ric(t, y) |: t ∈ (−∞, T ], dg(t)(x, y) ≤ r0 (the analogue of the constant C(x, r0) defined in (3.9)) is finite, and that N is simply-connected and has non-positive sectional curvatures. Then any ancient solution of sub-square-root growth u : (−∞, T ] × M → N to the harmonic map heat flow is constant. Remark 4.3. Theorem 4.1 should be compared with S.-Y. Cheng’s Liouville theorem [4] which gives an analogous statement for harmonic maps of sublinear growth when M is equipped with a fixed metric of non-negative Ricci curvature, see [33, Corollary 5.10] and [30] for stochastic proofs. All these proofs depend crucially on the Laplacian comparison theorem. In the case of backward super Ricci flow the stronger assumption of sub-square-root growth is needed because estimate (3.10) is weaker than estimate (3.11) which holds in the case of a fixed metric with non-negative Ricci curvature, see the discussion in Remark 3.2. H. Guo et al. / Stochastic Processes and their Applications 124 (2014) 3535–3552 3549 4.2. Space–time harmonic maps of bounded dilatation Let u : R+ × M → N be a space–time harmonic map. We say that u is of bounded dilatation if there is ...

Citation Context

... maps are characterized by the property that they map M-valued Brownian motions to N -valued martingales (see e.g. [11, Satz 7.157(ii)]), it is natural to study them using stochastic methods, ∗ Corresponding author. Tel.: +352 4666445867. E-mail addresses: guo@wzu.edu.cn (H. Guo), robert.philipowski@uni.lu (R. Philipowski), anton.thalmaier@uni.lu (A. Thalmaier). http://dx.doi.org/10.1016/j.spa.2014.06.004 0304-4149/ c 2014 Elsevier B.V. All rights reserved. 3536 H. Guo et al. / Stochastic Processes and their Applications 124 (2014) 3535–3552 and this has been done in a number of papers, e.g. [2,8,14,15,29,33]. In particular, stochastic representation formulae for the differential of harmonic maps have turned out to be a powerful tool to prove Liouville theorems, i.e. theorems stating that harmonic maps in a certain class of maps and under certain topological or geometric constraints are necessarily constant [33]. Due to Perelman’s proof of the geometrization and hence the Poincare conjecture using Ricci flow [25,27,26], there is now a strong interest in studying manifolds M with time-dependent geometry. In such a context, the notion of harmonic map turns out to be no longer appropriate; however, ...