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Colding Minicozzi Entropy in Hyperbolic Space

This note introduces a notion of entropy for submanifolds of hyperbolic space analogous to the one introduced by Colding and Minicozzi for submanifolds of Euclidean space. Several properties are proved for this quantity including monotonicity along mean curvature flow in low dimensions and a connection with the conformal volume.


INTRODUCTION
In [13], Colding and Minicozzi introduced the following functional on the space of n-dimensional submanifolds Σ ⊂ R n+k , Here, (4πt) − n 2 e − |x| 2 4t is the heat kernel of R n . Observe that in the definition of λ[Σ] this function is extended to all of R n+k in the obvious way. Colding and Minicozzi called this quantity the entropy of Σ and observed that, by Huisken's monotonicity formula [16], it is monotone along reasonable mean curvature flows. That is, if {Σ t } t∈[0,T ) is a family of complete submanifolds of polynomial volume growth that satisfy dx dt = H Σt , then λ[Σ t ] is monotone non-increasing in t. Here H Σt = ∆ Σt x = −H Σt n Σt is the mean curvature vector of Σ t and a submanifold, Σ, has polynomial volume growth if V ol R n+k (Σ ∩ B R ) ≤ M (1 + R) M for some M > 0. This notion of entropy has been extensively studied in recent years. See, for instance, [4-6, 8, 11, 12, 18, 24, 26]. In this short note, we introduce an analogous quantity for submanifolds of hyperbolic space, H n , and show that, at least in low dimensions, it is monotone along mean curvature flow. To that end, let H n (t, p; t 0 , p 0 ) be the heat kernel on H n with singularity at p = p 0 at time t = t 0 . That is, H n is the unique positive solution to ∂ ∂t − ∆ H n H n = 0 t > 0 lim t↓t0 H n = δ p0 .
In particular, Φ t0,p0 n restricts to the backwards heat kernel that becomes singular at (t 0 , p 0 ) on any totally geodesic H n ⊂ H n+k that goes through p 0 .
The author was partially supported by the NSF Grant DMS-1609340 and DMS-1904674. In analogy with the Euclidean setting, for any n-dimensional submanifold Σ ⊂ H n+k , define the hyperbolic entropy of Σ to be We remark that for any submanifold Σ ⊂ H n+k , λ H [Σ] ≥ 1 and if H n ⊂ H n+k is a totally geodesic copy of the n-dimensional hyperbolic plane, then λ H [H n ] = 1 -see Lemma A.1.
The main result of this paper is that, at least in sufficiently low dimensions, λ H is monotone non-increasing along reasonable mean curvature flows -i.e., flows of submanifolds of exponential volume growth. Here a submanifold Σ ⊂ H n+k has exponential volume growth if for some M > 0 and p 0 ∈ H n+k . In fact, at least in low dimensions, there is an analog of Huisken monotonicity.
and one has strict inequality unless Σ t is a minimal submanifold that is a cone over p 0 .
Here a submanifold, Σ, is a cone over a point p 0 , if, for every p ∈ Σ, the minimizing geodesic connecting p 0 to p in H n+k is contained in Σ. One readily checks that the only smooth n-dimensional cone over p 0 is a totally geodesic copy of H n containing p 0 . Remark 1.2. In fact, monotonicity holds provided the heat kernel of hyperbolic space, H n , satisfies the following convexity property: H n log H n (t, p; t 0 , p 0 ) − coth(ρ(p))∂ ρ log H n (t, p; t 0 , p 0 )g H n ≥ 0 for all p ∈ H n and t > t 0 . Here ρ(p) = dist H n (p, p 0 ) is the radial distance from p 0 . This is equivalent to ∂ 2 ρ log K n − coth(ρ)∂ ρ log K n ≥ 0. As the K n can be explicitly computed, one could, in principle, check this condition for any n and it seems reasonable to conjecture that the convexity, and hence Theorem 1.1, holds for N = ∞. However, the formulas become unwieldy for n ≥ 4 and so we do not pursue this. See [14] for an extensive study of properties of H n and K n .
An immediate consequence of this result, the classification of low entropy self-shrinkers in R 3 from [5] and the fact that the mean curvature flow of a closed hypersurface in hyperbolic space forms a singularity in finite time, is a sharp lower bound on the hyperbolic entropy of closed surfaces in H 3 . We remark that in his thesis [25], Zhu defines a notion of entropy, λ S [Σ], of an ndimensional submanifold, Σ, of S n+k that is monotone along mean curvature flow in the sphere. This is done by using the usual of embedding S n+k into R n+k+1 to think of Σ as a n-dimensional submanifold of R n+k+1 . He then defines the spherical entropy of Σ to be its usual entropy, λ[Σ], computed in the ambient Euclidean space. We also refer the reader to [23] where an entropy is defined in arbitrary (closed) ambient manifolds that is monotone when the metric has parallel Ricci curvature and non-negative sectional curvature. See also [20] where a notion of entropy in arbitrary Riemannian manifolds is obtained by isometrically embedding the ambient manifold in a large Euclidean space.
As a further application of Theorem 1.1, we consider its implications for complete minimal submanifolds in hyperbolic space. Specifically, we consider such minimal submanifolds that have a C 1 -regular asymptotic boundary on the ideal boundary, ∂ ∞ H n , of hyperbolic space. We remark that our computation seems to be related to the notion of holographic entanglement entropy introduced by Ryu and Takayanagi in [21,22] -see also [1,2] for a mathematical treatment closer to our own.
More precisely, recall that the Poincaré ball model of hyperbolic space H n is the unit ball B n = {x : |x| < 1} ⊂ R n with the Poincaré metric The ideal boundary, ∂ ∞ H n of H n is then identified with the sphere S n−1 = ∂B n . A ldimensional submanifold Σ ⊂ H n has C m -regular asymptotic boundary, provided that, after identifying with B n ,Σ ⊂B n is a C m -regular submanifold with boundary andΣ meets ∂B n = S n−1 orthogonally. For such a Σ, set ∂ ∞ Σ = ∂Σ =Σ ∩ S n−1 . Using the identification, we may think of ∂ ∞ Σ as a C m -regular (l − 1)-dimensional submanifold of ∂ ∞ H n .
As the isometries of H n correspond to Möbius transforms of B n , this means ∂ ∞ H n possesses a natural conformal structure, but not a unique Riemannian structure. In particular, there is not a well defined notion of volume for submanifolds Γ ⊂ ∂ ∞ H n . However, there is a well defined notion of conformal volume in the spirit of Li and Yau [19]. Specifically, given a l-dimensional submanifold Γ ⊂ ∂ ∞ H n , define the conformal volume of Γ to be Here Mob(S n−1 ). is the group of Möbius transformations of S n−1 and we have used the identification between S n−1 and ∂ ∞ H n . This is a well-defined geometric quantity and is essentially the (n − 1)-conformal volume of the embedding of Γ into S n−1 defined in [19]. Given a complete n-dimensional submanifold with C 1 -regular asymptotic boundary there is a natural relationship between its hyperbolic entropy and the conformal volume of its asymptotic boundary. .
If, in addition, Σ is minimal and n < N where N is given by Theorem 1.1

, then this inequality is an equality.
This is analogous to the relationship between the entropy of an asymptotically conical self-expander and the entropy of its asymptotic cone [7,Lemma 3.5]. Together with the main result [10] about topological properties of low entropy self-expanders, this suggests the following conjecture: A consequence of this conjecture and [3] and Σ ⊂ H 3 is a minimal surface with ∂ ∞ Σ = Γ, then Σ is a topological disk and, in particular, Γ is connected. The reason we restrict the conjecture to n = 2, is that the natural analog of relative expander entropy that is used in the analysis of [10] is the renormalized area (see [1,15]) and this quantity has special properties when n = 2. Nevertheless, it would be interesting see whether an analogous result held in higher dimensions -for instance it may only hold in odd ambient dimensions.

PROPERTIES OF THE HEAT KERNEL OF H n
We discuss here properties of the heat kernel on hyperbolic space that are relevant to the proof of Theorem 1.1 and Theorem 1.5. We make extensive use of the investigation of the heat kernel on hyperbolic space carried out by Davies and Mandouvalos in [14].
Recall, the symmetries of H n ensure that if H n (t, p; t 0 , p 0 ) is the heat kernel on H n with singularitity at p = p 0 and t = t 0 , then, there is a function K n so that For instance, as H 1 is just the Euclidean line, and this gives the heat kernel in one-dimension. Using this and (2.1), one obtains that gives the heat kernel on H 3 -see [14] for more details. We recall two recurrence relations for K n from [14]. The first is attributed to Millison: The second is: ds.
Using these relations, Davies and Mandouvalos obtained the following uniform estimates on the K n+1 [14, Thereom 3.1]: In particular, for any fixed p 0 ∈ H n+1 and R > 0, on has for t ≥ 1 then, for any m ≤ n, In particular, (2.5) holds for 1 ≤ m ≤ 3.
Proof. It is enough to show the result for m = n − 1 as the first claim will then follow by induction. Clearly, Moreover, one inequality is strict only if the other is. We prove the second inequality.
To that end, observe that combining (2.1) and (2.2) yields, As such, Moreover, this inequality is strict for ρ > 0 as can be seen by using the induction hypotheses and because sinh(ρ) > 0. Hence, the Cauchy-Schwarz inequality gives 2 and the inequality is strict when ρ > 0. That is, with strict inequality for ρ > 0 and the first claim is proved. For the second, observe Hence, and .

It follows that
One readily checks that ρ coth(ρ) − 1 ≥ 0 and the inequality is strict for The inequality follows. Likewise, using power series: Indeed, this is equivalent to Clearly, expanding as a power series gives One verifies that 8l 2 + 4l − 12 ≥ 0 for l ≥ 1 and so g(ρ) ≥ 0 as claimed. Hence, with strict inequality when ρ > 0. This completes the proof of the second claim.
The non-strict convexity of Proposition 2.1 is equivalent to a certain convexity property for the heat kernel of H n .

Proposition 2.2.
There is an N ∈ [4, ∞] so if n < N , then the heat kernel H n (t, p; t 0 , p 0 ) satisfies, for every t > t 0 , Proof. The metric g H n has the form g H n = dρ ⊗ dρ + sinh 2 (ρ)g S n−1 .

Hence, for any radial function
dρ ⊗ dρ and so the convexity property is equivalent to ∂ 2 ρ log K n − coth(ρ)∂ ρ K n ≥ 0. By Proposition 2.1, once this condition holds for N − 1 it holds for all n < N . Moreover, it holds for n = 3 and so we may take N ≥ 4.

PROOF OF THEOREM 1.1
In this section we prove the main montonicity result. First of all, consider hyperbolic space H n+k together with a distinguished point p 0 ∈ H n+k . Set ρ = dist H n+k (p, p 0 ). We use the following form of the metric in what follows Using this form of the hyperbolic metric, it is straightforward to compute that, for any C 2 function f = f (ρ(p)) on H n+k that is radial with respect to p 0 , so, in this case, Φ t0,p0 n is the backwards heat kernel on H n . For general k, taking the trace of (3.1) implies, We have the following analog of Huisken's formula: is a mean curvature flow in H n+k of n-dimensional submanifolds with exponential volume growth. For any t 0 ∈ (0, T ], p 0 ∈ H n+k and t ∈ (0, t 0 ) one has Here, n and {ν 1 (p), . . . , ν k (p)} is any choice of orthonormal basis of N p Σ t , the normal space of Σ t at p.

Proof. By the first variation formula we have
As Σ t has exponential volume growth, (2.3) and (2.1) can be used together with an integration by parts to see Standard geometric computations imply that on Σ t , where {ν 1 (p), . . . , ν k (p)} is an orthonormal basis of N p Σ t . Hence, Where we used that Hence, This completes the proof.
It is helpful to have the following elementary geometric lemma.
We are now ready to prove Theorem 1.1.
Proof of Theorem 1.1. As Φ t0,p0 n is radial with respect to p 0 , (3.1) implies that at p ∈ Σ t , with strict inequality unless ∇ ⊥ Σt ρ is identically zero. By Lemma 3.2, this occurs only if Σ t is a cone over p 0 .
Hence, by Proposition 3.1, d dt Σt Φ t0,p0 n (t, p)dV ol Σt ≤ 0 and the inequality is strict unless Σ t is a cone over p 0 . If Σ t is a cone over p 0 , then ∇ ⊥ Σt Φ t0,p0 n = 0 and so, by Proposition 3.1, In particular, the inequality is strict unless Σ t is a minimal submanifold that is a cone over p 0 . This completes the proof.
We remark that while we carried out these computations for smooth mean curvature flows, they carry over to weak flows in the sense of Brakke as in [17].

HYPERBOLIC ENTROPY OF COMPLETE MINIMAL SUBMANIFOLDS
In this section we estimate from below the hyperbolic entropy of complete submanifolds with C 1 -regular asymptotic boundary on the ideal boundary, ∂ ∞ H n , in terms of a certain boundary entropy -i.e., the conformal volume. These computations yield an exact formula when the submanifold is minimal of a dimension for which Theorem 1.1 holds.
Let us expand on the notation from the introduction. First, recall the Poincaré ball model of hyperbolic space, H n is the open unit ball in Euclidean space B n = {x : |x| < 1} ⊂ R n together with the Poincaré metric That is, for any model of hyperbolic space, (H n , g H n ), there is an isometry i : H n → B n so i * g P = g H n . The isometries of g P are given by the Möbius transforms of B n and so this identification is not unique. In fact, for any point p 0 ∈ H n , there is an isometry i : H n → B n so i(p 0 ) = 0. Moreover, if i, j : H n → B n satisfy i(p 0 ) = j(p 0 ) = 0, then i • j −1 is an orthogonal transformation of B n . In particular, in this case i * g R n = j * g R n while these metrics are different for identifications associated to distinct distinguished points. In what follows, we will always choose a distinguished point p 0 ∈ H n and an identification (i.e., an isometry) i : H n → B n with i(p 0 ) = 0. We use this identification to compactify H n and denote the ideal boundary of H n by ∂ ∞ H n which is identified with S n−1 = ∂B n in the natural way. Extend i :H n →B n in the obvious way. This compactification is independent, as a manifold with boundary, of the choice of p 0 and i.
A complete submanifold Σ ⊂ H n has C m -regular asymptotic boundary for 1 ≤ m ≤ ∞ if Σ ′ = i(Σ) ⊂ B n has the property that its closureΣ ′ is a C m -regular manifold with boundary, ∂Σ ′ ⊂ S n−1 = ∂B n andΣ ′ meets S n−1 orthogonally. Denote by ∂ ∞ Σ the submanifold corresponding to ∂Σ ′ in ∂ ∞ H n . As Möbius transformations are smooth and conformal this is a well defined notion independent of choice of identification.
Using the identification, i, ∂ ∞ H n has a well defined Riemannian metric induced from S n−1 = ∂B n . While this metric depends on p 0 , it is otherwise independent of the choice of isometry taking p 0 to 0. Let us denote this metric by g p0 ∂∞H n . Clearly, g p0 ∂∞H n and g q0 ∂∞H n are conformal for different choices of distinguished point p 0 and q 0 and so ∂ ∞ H n has a well defined conformal structure. In fact, the two metrics are related by a Möbius transform on the sphere. Fix a l-dimensional C m submanifold of ∂ ∞ H n and let i(Γ) ⊂ S n−1 be the corresponding submanifold of the sphere under the identification. Set If q 0 is a different choice of distinguished point, then, in general, it can happen that However, there is a Möbius transform, ψ ∈ Mob(S n−1 ) so that Hence, if we define the conformal volume of Γ ⊂ ∂ ∞ H n by then this is a quantity that is well defined independent of the choice distinguished point and of identification. Indeed, One has the following useful calculation that clarifies the meaning of V ol ∂∞H n+k (Γ) when Γ appears as the asymptotic boundary of a n-dimensional submanifold.
Lemma 4.1. Let Σ ⊂ H n+k be a n-dimensional submanfiold that has C 1 -regular asymptotic boundary. For any p 0 ∈ H n+k , Proof. Pick an identification i : H n+k → B n+k so that i(p 0 ) = 0. As i * g P = g H n+k , i is an isometry and so i(∂B H n+k r ) = ∂B gP r (0). Furthermore, as the conformal factor of g P is radial, ∂B gP r (0) = ∂B s (0) where r = ln 1+s 1−s . Set Σ r = Σ∩∂B H n+k r (p 0 ). and let Σ ′ = i(Σ). From the above, i(Σ r ) = Σ ′ ∩∂B s (0) = Σ ′ s . Let g r be the metric on Σ r induced from H n+k and g ′ s be the metric induced on Σ ′ s from g R n+k . Clearly, The definition of C 1 -regular asymptotic boundary ensures As s → 1, r → ∞ and so This completes the proof. Theorem 1.5 is a consequence of Theorem 1.1 and the following proposition: Proposition 4.2. Let Σ ⊂ H n+k be an n-dimensional submanifold with C 1 -regular asymptotic boundary. One has Observe that, by the definition of having a C 1 -regular asymptotic boundary there is an R 0 > 0 so that, for r ≥ R 0 , Σ meets ∂B H n+k r (p 0 ) transversally and so Σ r is a smooth (n − 1)-dimensional submanifold of ∂B H n+k r (p 0 ). It follows from Lemma 4.1 that, for any ǫ > 0, there is an R ǫ > R 0 so for r > R ǫ We also observe, that if ρ = dist H n+k (p, p 0 ), then |∇ H n+k ρ| = 1 for p = p 0 . It follows from the definition of having a C 1 -regular asymptotic boundary -specifically the fact that Σ is C 1 up to ∂ ∞ H n+k and meets this boundary orthogonally, that lim p→∞ |∇ Σ ρ| = 1.
Hence, by the co-area formula, for R > R ′ ǫ , one has and Now suppose for the moment that Σ = H n where H n is the totally geodesic hyperbolic plane going through p 0 . As Φ t0,p0 n (t, p) restricts to the backwards heat kernel on H n , one has, for all t < t 0 , The co-area formula and fact that H n is a cone over p 0 implies Here the final equality used that H n goes through p 0 . Hence, for any R > 0, .
It follows that, where the second equality again uses (2.4). Hence, we may send ǫ → 0 and obtain, In the exact same way, one obtains lim sup This proves that verifying the main claim.
We may now prove Theorem 1.5.
Proof of Theorem 1.5. By definition, for any fixed p 0 ∈ H n+k , Hence, Proposition 4.2 implies Taking the supremum over p 0 ∈ H n+k yields, .
This proves the first claim. To see the second claim, observe that if Σ is minimal, then we may think of Σ as a static solution of mean curvature flow. It readily follows from Lemma 4.1 that Σ has exponential volume growth. Hence, by Theorem 1.1, for all τ > 0 .
Taking the supremum over τ > 0 and p 0 ∈ H n+k yields .
Combined with the first claim this completes the proof.
Finally, it is enlightening to introduce the function One verifies that s is a boundary defining function and, in appropriate associated coordinates, the Poincaré metric has the form Hence, if i is an identification of H n+k with B n+k sending p 0 to 0 and σ = s • i, then σ 2 g H n+k is a conformal compactification in the sense of [15]. Suppose Σ ⊂ H n+k is a n-dimensional minimal submanifold that has C ∞ -regular asymptotic boundary. Using a boundary defining function like σ, Graham and Witten [15] showed that, when n is even, while if n is odd, then Here r(ǫ) = − ln ǫ 2 . They further showed that, when n is even, c 0 is independent of the choice of σ (and so independent of choice of p 0 ) and the same is true of d when n is odd. When n = 2, this quantity is precisely the renormalized area considered in [1]. This expansion is also the (Riemannian) analog of the entropy considered by Ryu and Takayanagi [21,22]. One observes that c 0 = V ol ∂∞H n+k (∂ ∞ Σ, p 0 ) so the conformal volume is related as the leading order behavior of this expansion. Comparing with [9] and [10] it seems the renormalized area is analogous to the relative entropy in some respects which is what motivates Conjecture 1.6 APPENDIX A. ELEMENTARY PROPERTIES OF λ H For the readers convenience, we collect here some elementary properties of the hyperbolic entropy.
Let H n ⊂ H n+k be a totally geodesic copy of hyperbolic space.
Proof. One clearly has λ H [H n ] ≥ 1 so we only have to prove the reverse inequality. To that end, by the Millison's identity (2.1) and the positivity of the heat kernel ∂ ρ K n = −e nt sinh(ρ)K n+2 ≤ 0 so Φ 0,p0 n is radially decreasing. Given p 0 ∈ H n ⊂ H n+k one clearly has H n Φ 0,p0 n (t, p)dV ol H n (p) = 1 while for p 0 ∈ H n , let p ′ 0 ∈ H n be the nearest point in H n to p 0 . For p ∈ H n it follows from the hyperbolic law of cosines that dist H n+k (p, p ′ 0 ) ≤ dist H n+k (p, p 0 ) and so for all p ∈ H n Φ 0,p0 n (t, p) = K n (t, dist H n+k (p, p 0 )) ≤ K n (t, dist H n+k (p, p ′ 0 )) = Φ 0,p ′ 0 n (t, p) and the claim follows.
Proof. Fix a point p 1 ∈ H n+k . As Σ is closed, there is an R 0 so Σ ⊂ B H n+k
As this is the supremem over a compact set of a continuous function the supremum is achieved and has a finite value.