## Final Report Summary - EGFLOW (Extrinsic Geometric Flows on Foliated Manifolds)

The research objective is to develop a new tool for studying Riemannian geometry of foliations: Extrinsic Geometric (EG) flows of metrics, recently introduced in [12] as deformations of Riemannian metrics on a manifold equipped with a codimension-one foliation. The specific objectives are as follows:

- Developing theory of parabolic EG flow for foliations of codimension-1 and of arbitrary codimension (existence and uniqueness theorems, converging as time variable runs to infinity, behavior of the curvature, singularities, geometry of limiting metrics etc), combining the EG flow approach with integral and variation formulae for foliations.

- Applications of EG flow to Gluck-Ziller, Walczak, and Toponogov problems: minimizing functions (like volume and energy) defined on foliated Riemannian manifolds, prescribing EG quantities of a foliation or its orthogonal distribution, foliated Riemannian submanifolds with given EG properties.

The project time was devoted to proving and publishing results (due to the project objectives), research in common with visiting scientists, training of the researcher (participation in seminars, conferences and reading literature) in areas of the project, organizing the International workshop, and supervising students, two of them defended their M.S. (S. Liflyandsky: Geometric flows on foliated manifold, and Y. Tenenhauzer: Nonlinear dynamics on a surface with a time-dependent metric).

Some novel concepts and tools of Riemannian geometry discussed in the project have been introduced by the researcher: variational formulas of EG quantities, the mixed Einstein-Hilbert action, the leaf-wise Schrodinger operator, the partial Ricci tensor and the partial Ricci flow. The results of the study could be relevant to geometers and young specialists in areas of Riemannian geometry and foliations, in EU (Spain, Poland, France), Japan, USA etc.

The summary overview of important goals:

1. Variation and integral formulas of EG quantities for a foliation.

By integral formula we mean the vanishing of integrals over a foliated manifold of expressions composed from the configuration tensors, the curvature tensor, and their scalar invariants. Variation formulas express variation of different geometrical quantities of a fixed foliation under truncated (i.e. along the orthogonal distribution D or the foliation F) variation of Riemannian metric of the manifold. These were generalized from a codimension one case ([10, 12]) to almost product structure manifolds and were used for two purposes: for minimization of actions and for development of EG flows.

2. Parabolic EG flows for a codimension one foliation.

The theory of hyperbolic (first order) EG flow for a codimension one foliation, see [12], has been extended (see [1, 9]) to the case of parabolic (second order) EG flow, when the existing tools (e.g. the maximum principle) allow us to prove convergence of solutions. This was based on the formulae for deformations of geometric quantities as the Riemannian metric varies along the leaves. Under certain assumptions, the existence/uniqueness of a solution and its convergence (as time variable runs to infinity) has been shown. Applications to prescribing higher mean curvatures (which are symmetric functions of the principal curvatures) of a foliation were given.

3. Minimization of the mixed Einstein-Hilbert action on a foliation.

This is imitative of Einstein-Hilbert action, when the scalar curvature is replaced by the mixed scalar curvature, Smix, which is the averaged mixed sectional curvature. In [3], the codimension one case, physical meaning and some solutions on globally hyperbolic space-times (foliated because of the smooth Geroch splitting) were discussed, and examples of solutions to the mixed field equations in empty space-time were obtained (provided the foliation is isoparametric and totally umbilical). In [15], the action was studied for arbitrary codimension foliation on a closed Riemannian manifold. We used integral formulas from the foliation theory ([2, 8]) and developed variational formulas for EG quantities of foliations, aiming to calculate the Euler-Lagrange equations (called the mixed field equations) of the action and find examples of solutions (e.g. twisted products metrics). We represented these equations in two equivalent forms: in terms of EG, and using the partial Ricci tensor. The partial Ricci curvature, r(X, X), is the mean value of sectional curvatures over all mixed planes containing a unit vector X in D.

4. D-conformal parabolic EG flows.

These were studied and applied to problems of prescribing the mean curvature of the orthogonal distribution D, and constancy of the mixed scalar curvature (i.e. the mixed Yamabe problem). In [4], we developed the D-conformal EG flow for arbitrary codimension foliations (the speed of the flow is proportional to the divergence of the mean curvature vector H). The flow is equivalent to the heat flow of 1-form dual to H, and under suitable assumptions, it was shown that there exists a solution converging to a metric with H = 0; actually under some topological assumptions we can prescribe the mean curvature. In [6, 11, 17, 18], we developed D-conformal EG flow, whose velocity along D is proportional to Smix, this has been used to examine the question: Which foliations admit a metric with a given property of Smix (e.g. positive/negative or constant)? The flow preserves harmonicity of foliations and yields the leaf-wise Burgers type equation for the mean curvature vector H of D. When H is leaf-wise conservative, its potential obeys the non-linear heat equation. Under certain conditions (in terms of spectral parameters of leaf-wise Schrodinger operator ? - b, where ? is the leaf-wise Laplacian), the flow admits a unique global solution, converging exponentially to a metrics, whose Smix is leaf-wise constant. Hence, in certain cases, there exists D-conformal metrics with Smix positive/negative or constant. In [14], authors study the mixed Yamabe problem (i.e. the constancy of Smix by a biconformal change of metrics) using a non-linear heat equation and spectral parameters of the leaf-wise Schrodinger operator.

5. The partial Ricci flow (first steps).

Since development of general EG flows for arbitrary codimension foliations is very complicated, we concentrated on special EG flows. The best example is the partial Ricci flow, dg/dt = - 2r(g), introduced and developed in [13] for a codimension one foliation. These results were generalized in [5] for arbitrary foliations: the leaf-wise Laplacian of the curvature tensor, the short-time existence and uniqueness, evolution of the curvature tensor, behavior of the flow on twisted products, examples to my hypothesis about convergence of the partial Ricci flow to a metric with a D-conformal partial Ricci tensor. This is related to Toponogov question about existence of totally geodesic foliations with positive mixed curvature. We expect that the partial Ricci flow might be useful to examine the question: When a foliation admits a metric with constant mixed sectional curvature?

6. Some results of the project were devoted to hypersurfaces in Euclidean space: extrinsic representation of the partial Ricci flow on rotational surfaces; convergence of the mean curvature flow of submanifolds in Euclidean and hyperbolic spaces with Gaussian density ([7]) that generalizes result for hypersurfaces in [J. of Evolution Equations, 10 (2010) 413-423], classification of hypersurfaces with codimension one totally geodesic foliations: they are an open dense subset either ruled, or surface-like, or partial tubes, or envelopes of a one-parameter families of flat hypersurfaces ([16]).

Project website: http://math.haifa.ac.il/ROVENSKI/rovenski_ERG.pdf.

- Developing theory of parabolic EG flow for foliations of codimension-1 and of arbitrary codimension (existence and uniqueness theorems, converging as time variable runs to infinity, behavior of the curvature, singularities, geometry of limiting metrics etc), combining the EG flow approach with integral and variation formulae for foliations.

- Applications of EG flow to Gluck-Ziller, Walczak, and Toponogov problems: minimizing functions (like volume and energy) defined on foliated Riemannian manifolds, prescribing EG quantities of a foliation or its orthogonal distribution, foliated Riemannian submanifolds with given EG properties.

The project time was devoted to proving and publishing results (due to the project objectives), research in common with visiting scientists, training of the researcher (participation in seminars, conferences and reading literature) in areas of the project, organizing the International workshop, and supervising students, two of them defended their M.S. (S. Liflyandsky: Geometric flows on foliated manifold, and Y. Tenenhauzer: Nonlinear dynamics on a surface with a time-dependent metric).

Some novel concepts and tools of Riemannian geometry discussed in the project have been introduced by the researcher: variational formulas of EG quantities, the mixed Einstein-Hilbert action, the leaf-wise Schrodinger operator, the partial Ricci tensor and the partial Ricci flow. The results of the study could be relevant to geometers and young specialists in areas of Riemannian geometry and foliations, in EU (Spain, Poland, France), Japan, USA etc.

The summary overview of important goals:

1. Variation and integral formulas of EG quantities for a foliation.

By integral formula we mean the vanishing of integrals over a foliated manifold of expressions composed from the configuration tensors, the curvature tensor, and their scalar invariants. Variation formulas express variation of different geometrical quantities of a fixed foliation under truncated (i.e. along the orthogonal distribution D or the foliation F) variation of Riemannian metric of the manifold. These were generalized from a codimension one case ([10, 12]) to almost product structure manifolds and were used for two purposes: for minimization of actions and for development of EG flows.

2. Parabolic EG flows for a codimension one foliation.

The theory of hyperbolic (first order) EG flow for a codimension one foliation, see [12], has been extended (see [1, 9]) to the case of parabolic (second order) EG flow, when the existing tools (e.g. the maximum principle) allow us to prove convergence of solutions. This was based on the formulae for deformations of geometric quantities as the Riemannian metric varies along the leaves. Under certain assumptions, the existence/uniqueness of a solution and its convergence (as time variable runs to infinity) has been shown. Applications to prescribing higher mean curvatures (which are symmetric functions of the principal curvatures) of a foliation were given.

3. Minimization of the mixed Einstein-Hilbert action on a foliation.

This is imitative of Einstein-Hilbert action, when the scalar curvature is replaced by the mixed scalar curvature, Smix, which is the averaged mixed sectional curvature. In [3], the codimension one case, physical meaning and some solutions on globally hyperbolic space-times (foliated because of the smooth Geroch splitting) were discussed, and examples of solutions to the mixed field equations in empty space-time were obtained (provided the foliation is isoparametric and totally umbilical). In [15], the action was studied for arbitrary codimension foliation on a closed Riemannian manifold. We used integral formulas from the foliation theory ([2, 8]) and developed variational formulas for EG quantities of foliations, aiming to calculate the Euler-Lagrange equations (called the mixed field equations) of the action and find examples of solutions (e.g. twisted products metrics). We represented these equations in two equivalent forms: in terms of EG, and using the partial Ricci tensor. The partial Ricci curvature, r(X, X), is the mean value of sectional curvatures over all mixed planes containing a unit vector X in D.

4. D-conformal parabolic EG flows.

These were studied and applied to problems of prescribing the mean curvature of the orthogonal distribution D, and constancy of the mixed scalar curvature (i.e. the mixed Yamabe problem). In [4], we developed the D-conformal EG flow for arbitrary codimension foliations (the speed of the flow is proportional to the divergence of the mean curvature vector H). The flow is equivalent to the heat flow of 1-form dual to H, and under suitable assumptions, it was shown that there exists a solution converging to a metric with H = 0; actually under some topological assumptions we can prescribe the mean curvature. In [6, 11, 17, 18], we developed D-conformal EG flow, whose velocity along D is proportional to Smix, this has been used to examine the question: Which foliations admit a metric with a given property of Smix (e.g. positive/negative or constant)? The flow preserves harmonicity of foliations and yields the leaf-wise Burgers type equation for the mean curvature vector H of D. When H is leaf-wise conservative, its potential obeys the non-linear heat equation. Under certain conditions (in terms of spectral parameters of leaf-wise Schrodinger operator ? - b, where ? is the leaf-wise Laplacian), the flow admits a unique global solution, converging exponentially to a metrics, whose Smix is leaf-wise constant. Hence, in certain cases, there exists D-conformal metrics with Smix positive/negative or constant. In [14], authors study the mixed Yamabe problem (i.e. the constancy of Smix by a biconformal change of metrics) using a non-linear heat equation and spectral parameters of the leaf-wise Schrodinger operator.

5. The partial Ricci flow (first steps).

Since development of general EG flows for arbitrary codimension foliations is very complicated, we concentrated on special EG flows. The best example is the partial Ricci flow, dg/dt = - 2r(g), introduced and developed in [13] for a codimension one foliation. These results were generalized in [5] for arbitrary foliations: the leaf-wise Laplacian of the curvature tensor, the short-time existence and uniqueness, evolution of the curvature tensor, behavior of the flow on twisted products, examples to my hypothesis about convergence of the partial Ricci flow to a metric with a D-conformal partial Ricci tensor. This is related to Toponogov question about existence of totally geodesic foliations with positive mixed curvature. We expect that the partial Ricci flow might be useful to examine the question: When a foliation admits a metric with constant mixed sectional curvature?

6. Some results of the project were devoted to hypersurfaces in Euclidean space: extrinsic representation of the partial Ricci flow on rotational surfaces; convergence of the mean curvature flow of submanifolds in Euclidean and hyperbolic spaces with Gaussian density ([7]) that generalizes result for hypersurfaces in [J. of Evolution Equations, 10 (2010) 413-423], classification of hypersurfaces with codimension one totally geodesic foliations: they are an open dense subset either ruled, or surface-like, or partial tubes, or envelopes of a one-parameter families of flat hypersurfaces ([16]).

Project website: http://math.haifa.ac.il/ROVENSKI/rovenski_ERG.pdf.