Constant heat flow and the isoparametric property Alessandro Savo - PowerPoint PPT Presentation

Obvious generalization: consider Clifford tori � S p ( t ) × S n − p ( 1 − t 2 ) for t ∈ (0 , 1). A Clifford torus bounds two domains in S n +1 , which are both harmonic.

Obvious generalization: consider Clifford tori � S p ( t ) × S n − p ( 1 − t 2 ) for t ∈ (0 , 1). A Clifford torus bounds two domains in S n +1 , which are both harmonic. But the sphere supports many more harmonic domains. Let us describe a larger class. This is due to Shklover (2000): Theorem Let Ω be a spherical domain bounded by a (connected) isoparametric hypersurface. Then Ω is harmonic. We will give another proof later.

Summary for constant curvature space forms We focused on Serrin’s overdetermined problem in the space forms R n , H n , S n :  ∆ u = 1 on Ω  u = 0 , ∂ u ∂ N = c on ∂ Ω 

Summary for constant curvature space forms We focused on Serrin’s overdetermined problem in the space forms R n , H n , S n :  ∆ u = 1 on Ω  u = 0 , ∂ u ∂ N = c on ∂ Ω  Domains which support a solution to this problem are called harmonic .

Summary for constant curvature space forms We focused on Serrin’s overdetermined problem in the space forms R n , H n , S n :  ∆ u = 1 on Ω  u = 0 , ∂ u ∂ N = c on ∂ Ω  Domains which support a solution to this problem are called harmonic . We have seen that in R n , H n , S n + (the hemisphere) the only harmonic domains are geodesic balls.

Summary for constant curvature space forms We focused on Serrin’s overdetermined problem in the space forms R n , H n , S n :  ∆ u = 1 on Ω  u = 0 , ∂ u ∂ N = c on ∂ Ω  Domains which support a solution to this problem are called harmonic . We have seen that in R n , H n , S n + (the hemisphere) the only harmonic domains are geodesic balls. In the whole sphere, we have many more harmonic domains, namely, domains bounded by an isoparametric hypersurface.

Summary for constant curvature space forms We focused on Serrin’s overdetermined problem in the space forms R n , H n , S n :  ∆ u = 1 on Ω  u = 0 , ∂ u ∂ N = c on ∂ Ω  Domains which support a solution to this problem are called harmonic . We have seen that in R n , H n , S n + (the hemisphere) the only harmonic domains are geodesic balls. In the whole sphere, we have many more harmonic domains, namely, domains bounded by an isoparametric hypersurface. Question : are there other examples or that is it ? In other words, is it true that the boundary of a harmonic spherical domain must be isoparametric ?

Summary for constant curvature space forms We focused on Serrin’s overdetermined problem in the space forms R n , H n , S n :  ∆ u = 1 on Ω  u = 0 , ∂ u ∂ N = c on ∂ Ω  Domains which support a solution to this problem are called harmonic . We have seen that in R n , H n , S n + (the hemisphere) the only harmonic domains are geodesic balls. In the whole sphere, we have many more harmonic domains, namely, domains bounded by an isoparametric hypersurface. Question : are there other examples or that is it ? In other words, is it true that the boundary of a harmonic spherical domain must be isoparametric ? Answer : no, it is not true: there are in fact other examples.

Even more exotic harmonic domains in spheres In a very recent paper, Fall, Minlend and Weth (2018) constructed new examples of harmonic domains in S n , by perturbing tubular neighborhoods of totally geodesic hypersurfaces.

Even more exotic harmonic domains in spheres In a very recent paper, Fall, Minlend and Weth (2018) constructed new examples of harmonic domains in S n , by perturbing tubular neighborhoods of totally geodesic hypersurfaces. Their boundaries are not isoparametric.

Even more exotic harmonic domains in spheres In a very recent paper, Fall, Minlend and Weth (2018) constructed new examples of harmonic domains in S n , by perturbing tubular neighborhoods of totally geodesic hypersurfaces. Their boundaries are not isoparametric. So, a classification seems to be at the moment out of reach, even in spaces as simple as the round sphere, which hosts a good variety of such domains.

Even more exotic harmonic domains in spheres In a very recent paper, Fall, Minlend and Weth (2018) constructed new examples of harmonic domains in S n , by perturbing tubular neighborhoods of totally geodesic hypersurfaces. Their boundaries are not isoparametric. So, a classification seems to be at the moment out of reach, even in spaces as simple as the round sphere, which hosts a good variety of such domains. Exception in dimension 2: a simply connected harmonic domain in S 2 must be a geodesic disk (Espinar-Mazet).

Isoparametric tubes We now isolate a class of Riemannian manifolds with boundary which always support a solution to the Serrin problem the isoparametric tubes.

Isoparametric tubes We now isolate a class of Riemannian manifolds with boundary which always support a solution to the Serrin problem the isoparametric tubes. By definition, the compact Riemannian manifold Ω n with smooth boundary ∂ Ω is called an isoparametric tube if there exists a smooth, compact submanifold P of Ω such that: a) Ω is a tube of radius R around P , that is Ω = { x : d ( x , P ) ≤ R }

Isoparametric tubes We now isolate a class of Riemannian manifolds with boundary which always support a solution to the Serrin problem the isoparametric tubes. By definition, the compact Riemannian manifold Ω n with smooth boundary ∂ Ω is called an isoparametric tube if there exists a smooth, compact submanifold P of Ω such that: a) Ω is a tube of radius R around P , that is Ω = { x : d ( x , P ) ≤ R } b) Each equidistant from P , say Σ t = { x ∈ Ω : d ( x , P ) = t } , t ∈ (0 , R ] , is a smooth hypersurface having constant mean curvature. • The submanifold P is called the soul of Ω, or focal submanifold of Ω. It can be shown that it is always minimal (Nomizu, Ge and Tang).

Examples It is by now classical that, in S n , any domain bounded by a connected • isoparametric hypersurface (that is, a hypersurface with constant principal curvatures) is an isoparametric tube. The soul is the focal submanifold of ∂ Ω. • Geodesic balls in constant curvature spaces (or, more generally, in a harmonic manifold) are isoparametric tubes. The soul is a point (the center of the ball). An annulus in R n is not an isoparametric tube. • • An isoparametric tube might have more than one boundary component: for example, a tubular neighborhood of an equator of the sphere. However, it cannot have more than two boundary components (otherwise it is not even a smooth tube). A solid revolution torus in R 3 is a smooth tube, but it is not • isoparametric : the soul is circle, and equidistant do not have constant mean curvature.

Theorem Any isoparametric tube is a harmonic domain.

Theorem Any isoparametric tube is a harmonic domain. Again, we have to show that the mean exit time function u : � ∆ u = 1 on Ω u = 0 on ∂ Ω has constant normal derivative.

Theorem Any isoparametric tube is a harmonic domain. Again, we have to show that the mean exit time function u : � ∆ u = 1 on Ω u = 0 on ∂ Ω has constant normal derivative. Radial functions. On any isoparametric tube we have the family of radial functions : these are the functions which are constant on every equidistant.

Theorem Any isoparametric tube is a harmonic domain. Again, we have to show that the mean exit time function u : � ∆ u = 1 on Ω u = 0 on ∂ Ω has constant normal derivative. Radial functions. On any isoparametric tube we have the family of radial functions : these are the functions which are constant on every equidistant. If we write ρ for the distance function to P : ρ ( x ) = d ( x , P ) then the function f on Ω is radial if and only if it can be expressed f = ψ ◦ ρ, for a smooth function ψ : [0 , R ] → R .

It is clear that a radial function has constant normal derivative on ∂ Ω: ∂ f ∂ N = �∇ f , N � = −�∇ f , ∇ ρ � = − ψ ′ ( R ) . Hence, to prove the theorem, it is enough to show that the torsion function is radial.

It is clear that a radial function has constant normal derivative on ∂ Ω: ∂ f ∂ N = �∇ f , N � = −�∇ f , ∇ ρ � = − ψ ′ ( R ) . Hence, to prove the theorem, it is enough to show that the torsion function is radial. Averaging operator (radialization) . We now define an operator A : C ∞ (Ω) → C ∞ (Ω) which will take a function to a radial function, called its radialization .

Recall that Ω is foliated by the equidistants (level sets of the distance function ρ to P ), hence any point x belongs to a unique equidistant; if x ∈ Ω \ P this will be the regular hypersurface Σ x = ρ − 1 ( ρ ( x )); and if x ∈ P then it will be simply P (the unique singular leaf).

Recall that Ω is foliated by the equidistants (level sets of the distance function ρ to P ), hence any point x belongs to a unique equidistant; if x ∈ Ω \ P this will be the regular hypersurface Σ x = ρ − 1 ( ρ ( x )); and if x ∈ P then it will be simply P (the unique singular leaf). Given f ∈ C ∞ (Ω) and x ∈ Ω \ P we define A f ( x ) as the average of f on the equidistant through x : 1 � A f ( x ) = f ; | Σ x | Σ x

Recall that Ω is foliated by the equidistants (level sets of the distance function ρ to P ), hence any point x belongs to a unique equidistant; if x ∈ Ω \ P this will be the regular hypersurface Σ x = ρ − 1 ( ρ ( x )); and if x ∈ P then it will be simply P (the unique singular leaf). Given f ∈ C ∞ (Ω) and x ∈ Ω \ P we define A f ( x ) as the average of f on the equidistant through x : 1 � A f ( x ) = f ; | Σ x | Σ x if x ∈ P we simply define A f ( x ) = 1 � f , | P | P where of course we use in both cases the measure induced by the Riemannian metric on Σ x and P , respectively.

If f ∈ C ∞ (Ω) its radialization A f has the following properties: A f is smooth as well; A f is radial; f is radial if and only if A f = f .

If f ∈ C ∞ (Ω) its radialization A f has the following properties: A f is smooth as well; A f is radial; f is radial if and only if A f = f . The crucial property of an isoparametric foliation is the constancy of the mean curvature of the leaves. This has the following important consequence.

If f ∈ C ∞ (Ω) its radialization A f has the following properties: A f is smooth as well; A f is radial; f is radial if and only if A f = f . The crucial property of an isoparametric foliation is the constancy of the mean curvature of the leaves. This has the following important consequence. The radialization commutes with the Laplacian: A ∆ f = ∆ A f . All the above facts are not difficult to prove (only a little technical at times).

The torsion function is radial . We can now prove that the torsion function u is radial; for that it is enough to show that A u = u .

The torsion function is radial . We can now prove that the torsion function u is radial; for that it is enough to show that A u = u . Let ˆ u = A u . Then, as the radialization commutes with the Laplacian: ∆ˆ u = ∆ A u = A ∆ u = A 1 = 1 , and of course ˆ u = 0 on ∂ Ω.

The torsion function is radial . We can now prove that the torsion function u is radial; for that it is enough to show that A u = u . Let ˆ u = A u . Then, as the radialization commutes with the Laplacian: ∆ˆ u = ∆ A u = A ∆ u = A 1 = 1 , and of course ˆ u = 0 on ∂ Ω. Then, u and ˆ u are two solutions of the boundary value problem � ∆ u = 1 u = 0 on ∂ Ω By uniqueness, they must coincide, hence A u = u , as asserted.

Heat equation: introduction Let M be a Riemannian manifold and k ( t , x , y ) be its heat kernel (solution of the heat equation having initial data given by the Dirac distribution at x ). The variable t > 0 is time, and x , y ∈ M .

Heat equation: introduction Let M be a Riemannian manifold and k ( t , x , y ) be its heat kernel (solution of the heat equation having initial data given by the Dirac distribution at x ). The variable t > 0 is time, and x , y ∈ M . k ( t , x , y ) is the temperature, at time t , at the point y , assuming that one unit of heat is placed at x at time t = 0.

Heat equation: introduction Let M be a Riemannian manifold and k ( t , x , y ) be its heat kernel (solution of the heat equation having initial data given by the Dirac distribution at x ). The variable t > 0 is time, and x , y ∈ M . k ( t , x , y ) is the temperature, at time t , at the point y , assuming that one unit of heat is placed at x at time t = 0. The heat kernel depends on the metric: t → 0 log k ( t , x , y ) = − d ( x , y ) 2 lim . 4

Heat equation: introduction Let M be a Riemannian manifold and k ( t , x , y ) be its heat kernel (solution of the heat equation having initial data given by the Dirac distribution at x ). The variable t > 0 is time, and x , y ∈ M . k ( t , x , y ) is the temperature, at time t , at the point y , assuming that one unit of heat is placed at x at time t = 0. The heat kernel depends on the metric: t → 0 log k ( t , x , y ) = − d ( x , y ) 2 lim . 4 One can measure distance using only a termometer.

Heat equation: introduction Let M be a Riemannian manifold and k ( t , x , y ) be its heat kernel (solution of the heat equation having initial data given by the Dirac distribution at x ). The variable t > 0 is time, and x , y ∈ M . k ( t , x , y ) is the temperature, at time t , at the point y , assuming that one unit of heat is placed at x at time t = 0. The heat kernel depends on the metric: t → 0 log k ( t , x , y ) = − d ( x , y ) 2 lim . 4 One can measure distance using only a termometer. There are deep intrisic relations between geometry and heat diffusion.

The heat equation Let Ω be a domain in a Riemannian manifold. Assume that at time t = 0 the temperature distribution is prescribed by a function φ 0 ( x ) on Ω, and that the boundary is kept at zero temperature at all times.

The heat equation Let Ω be a domain in a Riemannian manifold. Assume that at time t = 0 the temperature distribution is prescribed by a function φ 0 ( x ) on Ω, and that the boundary is kept at zero temperature at all times. If for example φ 0 is positive, heat will flow away from the domain because of refrigeration and eventually, at infinite time, the temperature will be constant, equal to zero, at all points of Ω.

The heat equation Let Ω be a domain in a Riemannian manifold. Assume that at time t = 0 the temperature distribution is prescribed by a function φ 0 ( x ) on Ω, and that the boundary is kept at zero temperature at all times. If for example φ 0 is positive, heat will flow away from the domain because of refrigeration and eventually, at infinite time, the temperature will be constant, equal to zero, at all points of Ω. But what is precisely the evolution of temperature ?

The heat equation Let Ω be a domain in a Riemannian manifold. Assume that at time t = 0 the temperature distribution is prescribed by a function φ 0 ( x ) on Ω, and that the boundary is kept at zero temperature at all times. If for example φ 0 is positive, heat will flow away from the domain because of refrigeration and eventually, at infinite time, the temperature will be constant, equal to zero, at all points of Ω. But what is precisely the evolution of temperature ? Denote by φ ( t , x ) the temperature at the point x ∈ Ω, at time t > 0. Then, φ ( t , x ) is a solution of the heat equation: ∆ φ ( t , x ) + ∂φ ∂ t ( t , x ) = 0 . where the Laplacian is acting on the space variable x .

In conclusion, the temperature function is a solution of the following initial-boundary value problem: ∆ φ ( t , x ) + ∂φ  ∂ t ( t , x ) = 0 for all x ∈ Ω , t > 0    φ (0 , x ) = φ 0 ( x ) for all x ∈ Ω    φ ( t , y ) = 0 for all y ∈ ∂ Ω and for all t > 0 In the last line we see the Dirichlet boundary condition, which correspond to boundary refrigeration at all times.

In conclusion, the temperature function is a solution of the following initial-boundary value problem: ∆ φ ( t , x ) + ∂φ  ∂ t ( t , x ) = 0 for all x ∈ Ω , t > 0    φ (0 , x ) = φ 0 ( x ) for all x ∈ Ω    φ ( t , y ) = 0 for all y ∈ ∂ Ω and for all t > 0 In the last line we see the Dirichlet boundary condition, which correspond to boundary refrigeration at all times. Indeed, once the initial data and the boundary conditions are chosen, the solution φ ( t , x ) exists and is unique .

In conclusion, the temperature function is a solution of the following initial-boundary value problem: ∆ φ ( t , x ) + ∂φ  ∂ t ( t , x ) = 0 for all x ∈ Ω , t > 0    φ (0 , x ) = φ 0 ( x ) for all x ∈ Ω    φ ( t , y ) = 0 for all y ∈ ∂ Ω and for all t > 0 In the last line we see the Dirichlet boundary condition, which correspond to boundary refrigeration at all times. Indeed, once the initial data and the boundary conditions are chosen, the solution φ ( t , x ) exists and is unique . Let us write φ t ( x ) . = φ ( t , x ). The maximum principle implies that, if φ 0 is a positive function, then φ t will be positive in the interior of Ω for all t > 0.

The constant flow property We now introduce an overdetermined problem for the heat equation. To isolate the geometric aspect of the problem, we will assume constant unit initial conditions.

The constant flow property We now introduce an overdetermined problem for the heat equation. To isolate the geometric aspect of the problem, we will assume constant unit initial conditions. Unit initial data . Let u = u ( t , x ) be the solution of the heat equation on Ω with initial data 1 and Dirichlet boundary conditions. If we write u t ( x ) . = u ( t , x ) this means that u satisfies: ∆ u t + ∂ u t  ∂ t = 0 on Ω    u 0 = 1    u t = 0 on ∂ Ω .

The constant flow property We now introduce an overdetermined problem for the heat equation. To isolate the geometric aspect of the problem, we will assume constant unit initial conditions. Unit initial data . Let u = u ( t , x ) be the solution of the heat equation on Ω with initial data 1 and Dirichlet boundary conditions. If we write u t ( x ) . = u ( t , x ) this means that u satisfies: ∆ u t + ∂ u t  ∂ t = 0 on Ω    u 0 = 1    u t = 0 on ∂ Ω . The physical interpretation is the following: u ( t , x ) is the temperature at time t > 0, at the point x ∈ Ω, assuming the initial temperature is uniformly constant, equal to 1, and that the boundary is subject to absolute refrigeration. � Note u ( t , x ) = Ω k ( t , x , y ) dy , where k ( t , x , y ) is the heat kernel of Ω.

By the maximum principle, u t is positive on the interior of Ω.

By the maximum principle, u t is positive on the interior of Ω. We now turn our attention to the heat content of Ω, which is the function of time: � H ( t ) = u t . Ω It is smooth on (0 , ∞ ) and is a decreasing function of t . In fact, H ′ ( t ) = d � � ∂ u t � � ∂ u t u t = ∂ t = − ∆ u t = − ∂ N . dt Ω Ω Ω ∂ Ω

By the maximum principle, u t is positive on the interior of Ω. We now turn our attention to the heat content of Ω, which is the function of time: � H ( t ) = u t . Ω It is smooth on (0 , ∞ ) and is a decreasing function of t . In fact, H ′ ( t ) = d � � ∂ u t � � ∂ u t u t = ∂ t = − ∆ u t = − ∂ N . dt Ω Ω Ω ∂ Ω Heat flow . The function ∂ u t ∂ N ( y ) gives, at any point y ∈ ∂ Ω, the pointwise heat flow at y . It is a smooth, positive function defined on ∂ Ω.

We expect that, for a general domain, the heat flow ∂ u t ∂ N is not constant on ∂ Ω.

We expect that, for a general domain, the heat flow ∂ u t ∂ N is not constant on ∂ Ω. Constant flow property . We say that Ω has the constant flow property if, for all fixed t > 0 , the normal derivative ∂ u t ∂ N is a constant function on ∂ Ω .

We expect that, for a general domain, the heat flow ∂ u t ∂ N is not constant on ∂ Ω. Constant flow property . We say that Ω has the constant flow property if, for all fixed t > 0 , the normal derivative ∂ u t ∂ N is a constant function on ∂ Ω . This additional request gives rise to an overdetermined problem, which can then be written: ∆ u t + ∂ u t  ∂ t = 0      u 0 = 1 on Ω (3)  u t = 0 , ∂ u t   ∂ N = ψ ( t ) on ∂ Ω for all t > 0   for a suitable smooth function ψ of the only variable t ∈ (0 , ∞ ).

Perfect heat diffusers Domains with the constant flow property are perfect heat diffusers in the following sense.

Perfect heat diffusers Domains with the constant flow property are perfect heat diffusers in the following sense. Given a smooth function φ ∈ C ∞ ( ∂ Ω), define φ t ( x ) . ˆ = ˆ φ ( t , x ) as the solution of the heat equation on Ω with boundary conditions prescribed by the function φ ( x ) (at all times) and zero initial conditions. (The boundary is a perfect heating-refrigerating machine).

Perfect heat diffusers Domains with the constant flow property are perfect heat diffusers in the following sense. Given a smooth function φ ∈ C ∞ ( ∂ Ω), define φ t ( x ) . ˆ = ˆ φ ( t , x ) as the solution of the heat equation on Ω with boundary conditions prescribed by the function φ ( x ) (at all times) and zero initial conditions. (The boundary is a perfect heating-refrigerating machine). That is, ˆ φ t is the unique solution of the problem: φ t + ∂ ˆ  φ t ∆ˆ ∂ t = 0     ˆ φ 0 = 0    ˆ  φ t = φ on ∂ Ω , for all t > 0

Let � ˆ H φ ( t ) = φ ( t , x ) dx Ω be the heat content at time t with boundary data φ . Clearly H φ (0) = 0.

Let � ˆ H φ ( t ) = φ ( t , x ) dx Ω be the heat content at time t with boundary data φ . Clearly H φ (0) = 0. Theorem Ω has the constant flow property if and only if H φ ( t ) = 0 for all t ≥ 0 and for all φ ∈ C ∞ 0 ( ∂ Ω) (smooth functions on ∂ Ω with zero mean). The theorem says in particular that if Ω has the constant flow property, and if the total boundary heat is zero at all times, then also the total inner heat content is identically zero at all times.

Let � ˆ H φ ( t ) = φ ( t , x ) dx Ω be the heat content at time t with boundary data φ . Clearly H φ (0) = 0. Theorem Ω has the constant flow property if and only if H φ ( t ) = 0 for all t ≥ 0 and for all φ ∈ C ∞ 0 ( ∂ Ω) (smooth functions on ∂ Ω with zero mean). The theorem says in particular that if Ω has the constant flow property, and if the total boundary heat is zero at all times, then also the total inner heat content is identically zero at all times. At x ∈ ∂ Ω the boundary acts as a refrigerator if φ ( x ) < 0, and acts as a heater if φ ( x ) > 0 (at least for small times).

Let � ˆ H φ ( t ) = φ ( t , x ) dx Ω be the heat content at time t with boundary data φ . Clearly H φ (0) = 0. Theorem Ω has the constant flow property if and only if H φ ( t ) = 0 for all t ≥ 0 and for all φ ∈ C ∞ 0 ( ∂ Ω) (smooth functions on ∂ Ω with zero mean). The theorem says in particular that if Ω has the constant flow property, and if the total boundary heat is zero at all times, then also the total inner heat content is identically zero at all times. At x ∈ ∂ Ω the boundary acts as a refrigerator if φ ( x ) < 0, and acts as a heater if φ ( x ) > 0 (at least for small times). Perfect heat diffusers . Then the constant flow property holds if and only if the incoming heat is perfectly balanced, at all times, by the outgoing heat , so that the total inner heat is preserved.

Constant flow implies harmonic It turns out that the constant flow property is stronger than harmonicity. Theorem Any domain with the constant flow property is also harmonic, that is, it supports a solution to the Serrin problem.

Constant flow implies harmonic It turns out that the constant flow property is stronger than harmonicity. Theorem Any domain with the constant flow property is also harmonic, that is, it supports a solution to the Serrin problem. This can be justified as follows. Introduce the function � ∞ v ( x ) = u ( t , x ) dt . 0 Note that the integral converges because u ( t , x ) decreases to zero exponentially fast as t → ∞ .

Constant flow implies harmonic It turns out that the constant flow property is stronger than harmonicity. Theorem Any domain with the constant flow property is also harmonic, that is, it supports a solution to the Serrin problem. This can be justified as follows. Introduce the function � ∞ v ( x ) = u ( t , x ) dt . 0 Note that the integral converges because u ( t , x ) decreases to zero exponentially fast as t → ∞ . One sees that ∆ v = 1 and moreover v = 0 on ∂ Ω. Hence v is the mean-exit time function of Ω.

Constant flow implies harmonic It turns out that the constant flow property is stronger than harmonicity. Theorem Any domain with the constant flow property is also harmonic, that is, it supports a solution to the Serrin problem. This can be justified as follows. Introduce the function � ∞ v ( x ) = u ( t , x ) dt . 0 Note that the integral converges because u ( t , x ) decreases to zero exponentially fast as t → ∞ . One sees that ∆ v = 1 and moreover v = 0 on ∂ Ω. Hence v is the mean-exit time function of Ω. If Ω has the CFP one sees that, differentiating in the normal direction, one has ∂ v ∂ N = c as well.

By Serrin’s result and the above, we have the following rigidity result: The only domains in R n , H n and S n + having the CFP are geodesic balls. We will see another, easier proof later.

By Serrin’s result and the above, we have the following rigidity result: The only domains in R n , H n and S n + having the CFP are geodesic balls. We will see another, easier proof later. As for the Serrin problem, In the whole sphere there are plenty of other interesting examples.

Every isoparametric tube has the constant flow property What about existence of domains with CFP ? Here considerations similar to the Serrin problem apply.

Every isoparametric tube has the constant flow property What about existence of domains with CFP ? Here considerations similar to the Serrin problem apply. Theorem Any isoparametric tube has the constant flow property.

Every isoparametric tube has the constant flow property What about existence of domains with CFP ? Here considerations similar to the Serrin problem apply. Theorem Any isoparametric tube has the constant flow property. For the proof, recall that the radialization A commutes with the Laplacian. To show that the temperature function u t has constant normal derivative (for all t ), it is enough to show that it is radial, or that: A u t = u t .

Every isoparametric tube has the constant flow property What about existence of domains with CFP ? Here considerations similar to the Serrin problem apply. Theorem Any isoparametric tube has the constant flow property. For the proof, recall that the radialization A commutes with the Laplacian. To show that the temperature function u t has constant normal derivative (for all t ), it is enough to show that it is radial, or that: A u t = u t . The argument is the same as before: the function A u t is still a solution of the heat equation, with initial condition equal to A u 0 = 1 and, obviously, Dirichlet boundary conditions.

Every isoparametric tube has the constant flow property What about existence of domains with CFP ? Here considerations similar to the Serrin problem apply. Theorem Any isoparametric tube has the constant flow property. For the proof, recall that the radialization A commutes with the Laplacian. To show that the temperature function u t has constant normal derivative (for all t ), it is enough to show that it is radial, or that: A u t = u t . The argument is the same as before: the function A u t is still a solution of the heat equation, with initial condition equal to A u 0 = 1 and, obviously, Dirichlet boundary conditions. As the initial and boundary values data of A u t are the same as those of u t , we have by uniqueness A u t = u t .

Geometric rigidity So far, we have isolated a whole class of manifolds with the constant flow property, the isoparametric tubes. We can ask if there are other ”exotic” examples, like those constructed by Fall, Minlend and Weth for the Serrin problem.

Geometric rigidity So far, we have isolated a whole class of manifolds with the constant flow property, the isoparametric tubes. We can ask if there are other ”exotic” examples, like those constructed by Fall, Minlend and Weth for the Serrin problem. We prove that, for analytic metrics, isoparametricity is also a necessary condition for having the constant flow property. Here is the main result.

Geometric rigidity So far, we have isolated a whole class of manifolds with the constant flow property, the isoparametric tubes. We can ask if there are other ”exotic” examples, like those constructed by Fall, Minlend and Weth for the Serrin problem. We prove that, for analytic metrics, isoparametricity is also a necessary condition for having the constant flow property. Here is the main result. Theorem (S. 2017) Let Ω be a compact analytic manifold with smooth boundary. Assume that it has the constant flow property. Then Ω is an isoparametric tube around a minimal submanifold of M.

Geometric rigidity So far, we have isolated a whole class of manifolds with the constant flow property, the isoparametric tubes. We can ask if there are other ”exotic” examples, like those constructed by Fall, Minlend and Weth for the Serrin problem. We prove that, for analytic metrics, isoparametricity is also a necessary condition for having the constant flow property. Here is the main result. Theorem (S. 2017) Let Ω be a compact analytic manifold with smooth boundary. Assume that it has the constant flow property. Then Ω is an isoparametric tube around a minimal submanifold of M. Then, we have a complete characterization, in the analytic case, of the class of domains with the constant flow property: this class coincides with the class of isoparametric tubes.

Geometric rigidity So far, we have isolated a whole class of manifolds with the constant flow property, the isoparametric tubes. We can ask if there are other ”exotic” examples, like those constructed by Fall, Minlend and Weth for the Serrin problem. We prove that, for analytic metrics, isoparametricity is also a necessary condition for having the constant flow property. Here is the main result. Theorem (S. 2017) Let Ω be a compact analytic manifold with smooth boundary. Assume that it has the constant flow property. Then Ω is an isoparametric tube around a minimal submanifold of M. Then, we have a complete characterization, in the analytic case, of the class of domains with the constant flow property: this class coincides with the class of isoparametric tubes. This also gives an analytic characterization of the isoparametric condition.

Proof: main steps We have a domain with the constant flow property, and we must show that it is an isoparametric tube. In particular, we have to find out what is the soul of it (focal submanifold).

Proof: main steps We have a domain with the constant flow property, and we must show that it is an isoparametric tube. In particular, we have to find out what is the soul of it (focal submanifold). At the moment we have, at our disposal, only the boundary of Ω.

Proof: main steps We have a domain with the constant flow property, and we must show that it is an isoparametric tube. In particular, we have to find out what is the soul of it (focal submanifold). At the moment we have, at our disposal, only the boundary of Ω. Step 1. The mean curvature is constant on each equidistant. Define a function η in a neighborhood of ∂ Ω as follows: η ( x ) = mean curvature of Σ x at x where Σ x is the equidistant to the boundary containing x .

Proof: main steps We have a domain with the constant flow property, and we must show that it is an isoparametric tube. In particular, we have to find out what is the soul of it (focal submanifold). At the moment we have, at our disposal, only the boundary of Ω. Step 1. The mean curvature is constant on each equidistant. Define a function η in a neighborhood of ∂ Ω as follows: η ( x ) = mean curvature of Σ x at x where Σ x is the equidistant to the boundary containing x . In fact, η = ∆ ρ where ρ : Ω → R is the distance function to the boundary.

Proof: main steps We have a domain with the constant flow property, and we must show that it is an isoparametric tube. In particular, we have to find out what is the soul of it (focal submanifold). At the moment we have, at our disposal, only the boundary of Ω. Step 1. The mean curvature is constant on each equidistant. Define a function η in a neighborhood of ∂ Ω as follows: η ( x ) = mean curvature of Σ x at x where Σ x is the equidistant to the boundary containing x . In fact, η = ∆ ρ where ρ : Ω → R is the distance function to the boundary. The aim in the first step is to show that η is radial, that is, it is constant on equidistants to the boundary.

Main technical step This is achieved by the following theorem.