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Outline of today’s lecture 2/ 20

1. Convex geometry in Euclidean space.

2. Geometry of horospherical convex hypersurface in Hyperbolic space.

3. New quermassintegrals, curvature flows, geometric inequalities, some remarks.

Convex geometry in Euclidean space 3/ 20Let ⌦ be a convex domain in R

n+1. A closed Halfspace H is said to support ⌦ (at p)if ⌦ ⇢ H and p 2 @⌦ \ @H. The boundary of H is called a supporting hyperplane.The closure of ⌦ is the intersection of its supporting hyperplanes.

Rn+1

⌦o

p

@H

Support function of ⌦ is a function s : Sn ! R

s(z) = supx2⌦

hx, zi, z 2 Sn,

which completely determines ⌦

⌦ =\

z2Sn

ny 2 R

n+1 : hy, zi < s(z)o.

t

sf

e

Convex geometry in Euclidean space 4/ 20

The boundary ⌃ = @⌦ of a smooth bounded convex domain is a convexhypersurface in R

n+1. Let⌫ : ⌃ ! S

n

be the Gauss map of ⌃ = @⌦. When ⌃ is strictly convex (i > 0), the Weingartenoperator W = d⌫ is positive definite. Then ⌫ : ⌃ ! S

n is a diffeomorphism.

LemmaLet ⌃ = @⌦ be a strictly convex hypersurface in R

n+1. Then ⌫⇤gSn = h

2.

0mr

Kiso

proof µ gsn 9,13 39gal hi 3 hiIfehikhjk hi ij

Convex geometry in Euclidean space 5/ 20The support function s : Sn ! R of a convex hypersurface ⌃ is defined by

s(z) =h⌫�1(z), zi,

where z 2 Sn is the outer unit normal at the point ⌫�1(z) 2 ⌃.

LemmaLet ⌃ = @⌦ be a strictly convex hypersurface in R

n+1. Then

⌫�1(z) = s(z)z + rs(z).

The second fundamental form h and support function are related by

⌫⇤⇣r2

s + sg

⌘= h.

In other words, we can reparametrize a strictly convex ⌃ as an embedding

X : Sn ! R

n+1

z 7! s(z)z + rs(z)

b 2 Sh

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hij I D salt 51245g

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5ij 1h4

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g ik A Tig w 1i

eigenvalues y my f T Inprincipal curvatures ri Yki it n

Convex geometry in Euclidean space 6/ 20

Denote A[s] = r2s + sg. Then

⌫⇤gSn =h

2, ⌫⇤A[s] = h

It follows that the eigenvalues r1, · · · , rn of A[s] with respect to g = gSn are the sameas the eigenvalues of h with respect to h

2, which are 1/1, · · · , 1/n. That is,

ri =1i

, i = 1, · · · , n

are the principal curvature radii.

LemmaLet X : M ! R

n+1 be a strictly convex solution of

@@t

X = �F()⌫.

The flow is equivalent to the evolution of the support function of ⌃t = X(M, t)

@@t

s(z, t) = �F(A[s]�1), on Sn ⇥ [0, T).

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Hyperbolic case 7/ 20

We now describe an analogue geometry in hyperbolic space Hn+1.

I Supporting hyperplanes in Euclidean space are replaced by supportinghorospheres in H

n+1, which are hypersurfaces with constant principal curvatures

i = 1, i = 1, · · · , n

everywhere.

I A domain ⌦ in Hn+1 is called “horospherically convex” (or “h-convex”) for

each p 2 @⌦, there is a horosphere enclosing ⌦ and touching at p. If @⌦ issmooth, “h-convexity” is equivalent to that principal curvatures of @⌦ satisfyi > 1, i = 1, · · · , n.

sectional curvatureo

strictly h convex Eki I

Horospheres 8 / 20

Rn+1 ⇥ {0}

Hn+1

In Ponincaré disc model, given each point z 2 Sn

1 there exists a family ofHorospheres touching at z and foliating the whole space.

In hyperboloid model, given a point z 2 Sn, the horospheres touching at z are given

byHz(s) = {X : |X|2 = �1, hX, (z, 1)i = �es}, s 2 R

where s is the signed distance to the north pole N = (0, 1). Denote Bz(s) thehoro-balls enclosed by horospheres Hz(s).

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OO

pit HE Xo't

IHH 1 12 1 Xo o

iiin'I normal ri

X T E 11H U ETE Du X E ft W U o

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X IX T I Xi IZ 1 X t

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sinks

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s

i Given zesn a family of horospheresHats f Xt IH't X Iz D es

horo balls i.BZ s1 fxe1Hntt o x Iz 1 es

Support function 9/ 20

For each z 2 Sn we define the horospherical support function of an h-convex ⌦ by

u(z) := inf{s 2 R : ⌦ ⇢ Bz(s)} = sup{log(�X · (z, 1)), X 2 @⌦}.

Hn+1

⌦z 2 S

n

1 = @Hn+1

p

The horospherical support function completely determines an h-convex domain ⌦:

⌦ =\

z2Sn

Bz(u(z)).

8x9

Recovering the region from support function 10 / 20The horospherical Gauss map e : ⌃ ! S

n:

e(p) = ⇡(X(p)� ⌫(p)) 2 Sn

where ⇡(x, y) = y/x is the radial projection from the light cone to Sn ⇥ {1}.

If i > 1 everywhere,

De(v) = D⇡��

X�⌫((W � I)(v)) , v 2 T⌃,

it follows that e is a diffeomorphism from ⌃ to Sn, and we can reparametrize an

h-convex hypersurface using e�1.

Proposition (Andrews-Chen-W., 2018)

We can reparametrize an h-convex ⌃ = @⌦ using the horospherical support functionu by X = X(e�1(·)) : Sn ! H

n+1,

X(z) =

✓�e

uru +�1

2e

u|ru|2 � sinh u�z,

12

eu|ru|2 + cosh u

◆.

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Recovering the region from support function 11/ 20If i > 1 everywhere, for each z 2 S

n there is a unique point of ⌃ touching@(Bz(u(z))), which we label as X(z).

Choose local coordinates {xi} for Sn near z. We write X(z) as a linear combination of

the basis consisting of the two null elements (z, 1) and (�z, 1), together with (zj, 0),where zj = @z

@xj for j = 1, · · · , n:

X(z) = ↵(�z, 1) + �(z, 1) + � j(zj, 0)

for some coefficients ↵, �, � j. The idea is to find the coefficients using the facts

I X(z) 2 Hn+1.

I X(z) · (z, 1) = �eu(z) since X(z) 2 @Bz(u(z)).

I The normal to @⌦ coincides with the normal to the horosphere @Bz(u(z)).

⌫ = X(z)� e�u(z)(z, 1).

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ri eat Ui

A condition for h-convexity 12/ 20Given a function u 2 C

1(Sn), we can use the expression

X(z) =��euru, 0

�+

12(eu|ru|2 + e�u)(z, 1) +

12

eu(�z, 1)

to define a map to hyperbolic space. The unit normal is given by ⌫ = X � e�u(z, 1).Differentiation in z gives

(W � I)(DiX) = uie�u(z, 1)� e�u(@iz, 0).

Taking the inner product with �(@jz, 0) gives

(W � I)k

i Akj = e�ugij

where

Akj = �h@kX, (@jz, 0)i = rkrjeu � 12

eu|ru|2gkj + sinh ugkj .

Proposition (Andrews-Chen-W., 2018)

The map X : Sn ! Hn+1 defined in terms of a function u 2 C

1(Sn) is an embeddingof an h-convex hypersurface if and only if the matrix Akj[u] is positive definite.

Ii Sn IHn11

O

O

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ri L hyperbolic principal radiiki l

ppht Xo XoA

coshrtzkten hint't WhIhrawhere HH dist Itt N

IHH

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Remarks 13 / 201. This motivates us to define the hyperbolic principal curvature radii as:

ri =1

i � 1, i = 1, · · · , n.

A similar development was presented by Espinar-Gálvez-Mira (2009), but in aslightly different context: In that paper the ‘horospherically convex’ regions arethose which are intersections of complements of horo-balls (corresponding toprincipal curvatures greater than �1 everywhere, while we deal with regionswhich are intersections of horo-balls, corresponding to principal curvaturesgreater than 1. Our condition is more stringent but is more useful for theevolution equations we consider later.

2. Let r(z) = distHn+1(o, X(z)). Then

cosh r(z) =12

eu|ru|2 + cosh u

Differentiation gives sinh r(z)ri = Aij[u]uj. It follows rr = 0 iff ru = 0, andmaxSn r = maxSn u.

Ki l

ki I

Remarks 14 / 203. We can ask similar questions that have been studied in Euclidean convex

geometry: for example, the Christoffel-Minkowski problem in hyperbolicspace, prescribed curvature problem in hyperbolic space, etc.

4. We can study the geometric flows by smooth functions of the hyperbolicprincipal curvature radii (equivalently, of the shifted principal curvaturei = i � 1), and study the geometric inequalities for h-convex hypersurfaces.

LemmaLet X : M ⇥ [0, T) ! H

n+1 be a smooth h-convex solution to the flow

@@t

X = �F(� 1)⌫.

in Hn+1. Then it is equivalent to the following initial value problem

8<

:

@@t

u = � F(e�uA[u]�1),

u(·, 0) = u0(·)

on Sn ⇥ [0, T), where u is the horospherical support function of ⌃t = X(M, t).

l it n

K Kii kn

principal curvatures

on Smx 6 1

00

New Quermassintegrals 15/ 20We next discuss some results on curvature flows by shifted principal curvatures.

We define a new family of ‘modified Quermassintegrals’ in hyperbolic space, whichare natural under the condition of h-convexity:

eWk(⌦) =kX

i=0

(�1)k�i

k

i

!Wi(⌦), k = 0, · · · , n.

These are characterised by their variation equation: If @X

@t= F⌫, then

d

dt

eWk(⌦t) =

Z

⌃t

FEk dµt,

where Ek = Ek() is the kth elementary symmetric function of the ‘shifted’ principalcurvatures i = i � 1. Denote fk(r) = eWk(B(r)). We proved

Theorem (Andrews-Chen-W. 2018)Let ⌦ ⇢ H

n+1 be a smooth, bounded and h-convex domain. Then

eWk(⌦) � fk � f�1` (eW`(⌦)), 0 ` < k n (1)

with equality holding if and only if ⌦ is a geodesic ball.

K lk kn

ex.D z Y il Eit

O

FINI WTCBM

New Quermassintegrals 16/ 20

1. The inequalities (1) are new and can be viewed as an improvement ofWang-Xia’s (2014) result.

2. The proof is by applying the following Quermassintegral preserving flow

@@t

X =

�(t)�

✓Ek

E`

◆ 1k�`

!⌫, 0 ` < k n

with �(t) chosen to keep eW`(⌦t) constant.3. Complication: If k > n/2, then limr!1 fk(r) < 1. To make sense of the

Theorem we first need to prove that Wk(⌦) is in the range of fk.

We prove: Wk is monotone with respect to inclusion for h-convex domains, i.e.if ⌃i = @⌦i for i = 1, 2 are h-convex and ⌦1 ⇢ ⌦2, then Wk(⌦1) Wk(⌦2).The fact that Wk(⌦) < limr!1 fk(r) follows since we can always enclose ⌦ bya large sphere.

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Inverse mean curvature type flow 17/ 20

We also studied the following inverse mean curvature type flow (with Xianfeng Wangand Tailong Zhou (2019))

@@t

X =1

H � n⌫

for h-convex hypersurfaces in hyperbolic space.1. The solution ⌃t expands to infinity in finite time, and is asymptotically round in

the sense that the oscillation decays to zero exponentially.

2. This is in contrast to the asymptotical behavior of IMCF

@@t

X =1H⌫

in hyperbolic space, as André Neves (2010), and P.-K. Hung and M.-T. Wang(2015) constructed examples to show the limiting shape of IMCF in hyperbolicspace is not necessarily round.

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n

in 1H's

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Asymptoticallyhyperbolic

Locally constrained curvature flow 18/ 20With Haizhong Li and Yingxiang Hu (2020), we introduced a new locallyconstrained curvature flow

@@t

X =

✓(cosh r � hsinh r@r, ⌫i)

Ek�1

Ek

� hsinh r@r, ⌫i◆⌫

for hypersurfaces in hyperbolic space.

1. The convergence to a geodesic sphere can be proved for h-convex hypersurfacesin hyperbolic space.

2. The flow preserves Wk and decreases Wk+1.

3. New optimal geometric inequalitiesZ

⌃

uLk � cn,k|⌃|n+1�2k

n

for h-convex hypersurfaces. The case k = 1 was proved by Brendle, Hung andWang (2012) for mean convex and star-shaped hypersurfaces.

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Concluding remarks 19/ 20In this course, we discussed the hypersurface curvature flows in Euclidean space andin hyperbolic space, with focus on their applications in geometric inequalities:

Isoperimetric inequailty, Alexandrov � Fenchel inequalities.

There are many other interesting topics on hypersurface flows:1. Mean curvature flow and fully nonlinear contracting curvature flows:

Singularity analysis and application in topology.

2. Curvature flows in general ambient space, and for hypersurfaces with boundary.

3. Application in convex geometry.

4.

Other geometric flows: Ricci flow, Kähler Ricci flow, Harmonic map heat flow,Yang-Mills flow, G2-Laplacian flow, etc. They are (degenerate) parabolic equations /systems of certain geometric structures on a Riemannian manifold. The mainmotivation is the application in geometric and topology of the underlying manifold.

T ScheuerSH Warped product G Want

in RM C Xia

non compact in 1H

Thank you!

- Q & A.

-eáµyongwei@ustc.edu.cn- http://staff.ustc.edu.cn/~yongwei/