2.2 Curvature functions
2.2.5 Examples
We now describe some examples of curvature functions which define admissible flow speeds, and discuss subsets of these which satisfy each of the auxiliary conditions. The cases for which no proof or reference is given are easily checked.
Let us first recall that theelementary symmetric polynomials (in n-variables) are the functions Sk:Rn→R,k= 0, . . . , ndefined by Sk(z1, . . . , zn) := n k −1 X 1<i1<···≤ik≤n zi1. . . zik, fork= 1, . . . , n , S0(z1, . . . , zn) := 1.
We note that, along an immersion, the elementary symmetric polynomials give rise to several well-known curvature invariants, such as the mean curvature, H = nS1(~κ), the
scalar curvature, Scal =n(n−1)S2(~κ), and the Gauß curvature, K=Sn(~κ).
Example 2.2 (Admissible flow speeds). The following symmetric functions define admissi- ble flow speeds:
1. The curve shortening flow: Up to a rescaling of the time parameter, the only ad- missible flow speed for the flow (CF) in one space dimension is f(z) = z. The corresponding flow is called thecurve shortening flow.
2. The arithmetic mean,
f(z) =S1(z),
4
defines an admissible flow speed on all of Rn. It is positive on the positive mean half-space, Γ1 :={z∈Rn:z1+· · ·+zn>0}. The corresponding curvature function
is the (normalized) mean curvature and the corresponding flow is (up to a rescaling of the time parameter) the well-known mean curvature flow.
3. The power means,
Hr(z) := 1 n n X i=1 zir !1r , ifr6= 0, H0(z) := n Y i=1 zi !n1 ,
define positive admissible flow speeds on the cone Γr := {z ∈ Rn : Pni=1zir >
0, zir−1>0 for each i}. Note that Γrcontains the positive cone Γ+:={z∈Rn:zi>
0 for each i}. The corresponding curvature functions include (up to normalization) the mean curvature (r= 1), the harmonic mean curvature (r=−1), the magnitude of the second fundamental form (r = 2), and the n-th root of the Gauß curvature (r = 0).
4. Ratios of consecutive elementary symmetric polynomials: The functions
f = Sk
Sk−1
, k= 1, . . . , n
define positive admissible speeds on the cone Γk := {z ∈ Rn : Sl(z) > 0 for l ≤ k} (see, for example, Lieberman 1996, Chapter XV). Note that Γk contains the
positive cone Γ+; in fact, Γ1 ⊃ · · · ⊃ Γn = Γ+ (see, for example, Huisken and
Sinestrari 1999a, Proposition 2.6). The corresponding curvature functions include (up to normalization) the mean curvature (n= 1) and the harmonic mean curvature (k=n).
5. Roots of the elementary symmetric polynomials: The functions
f =S 1
k
k , k= 1, . . . , n
define positive admissible speeds on the cone Γk := {z ∈ Rn : Sl(z) > 0 for l ≤ k} (see, for example, Lieberman 1996, Chapter XV, or Example 8 below). The corresponding curvature functions include (up to normalization) the mean curvature (n = 1), the square root of the scalar curvature (n = 2), and the n-th root of the Gauß curvature (k=n).
6. Positive linear combinations
f =X
i
of (positive) admissible flow speeds fi : Γ → R define (positive) admissible flow speeds f : Γ→R.
7. Weighted geometric means
f =
N
Y
i=1
fωi
i such thatωi ≥0 for each iand N X i=1 ωi= 1 ! ,
of positive admissible flow speeds fi : Γ→ Rdefine positive admissible flow speeds
f : Γ→R.
8. Roots of ratios of elementary symmetric polynomials: The function
f = Sk Sl 1 k−l , 0≤l < k≤n ,
is the geometric mean of fi = SSi−i1 fori=l+ 1, . . . , k, and hence defines a positive
admissible speed function on the cone Γk:={z∈Rn:Si(z)>0 for each i≤k}.
9. Homogeneous functions of admissible speeds: Iffi : Γ→R,i= 1, . . . , N are admis-
sible speeds and φ : ⊕N
i=1fi(Γ) ⊂ RN → R is a smooth, degree one homogeneous (positive) function which is monotone increasing in each variable, then
f :=φ(f1, . . . , fN)
is a (positive) admissible speed.
We next consider flow speeds which satisfy one of the auxiliary conditions. Surface flows
Example 2.3 (Admissible surface flows). The following symmetric functions define admis- sible speeds for surface flows:
1. Admissible speeds: All of the examples from Example 2.2 (withn= 2).
2. A general construction for positive admissible speeds: Write z1, z2 in polar coordi-
nates (r, θ) with angle measured anti-clockwise from the positive ray; that is,
r= q z2 1 +z22, cosθ= z1+z2 p 2(z12+z22), sinθ= z2−z1 p 2(z21+z22).
Then, writing f =rφ(θ) for some φ: (−θ0, θ0) → R, Conditions 1 (i)–(iv) become:
θ0<3π/4,φ >0, and
A(θ)< φ
0(θ)
where A(θ) := −∞, θ∈(−3π/4,−π/4] ; cosθ−sinθ cosθ+sinθ, θ∈(−π/4,3π/4) ; and B(θ) := cosθ+sinθ sinθ−cosθ, θ∈(−3π/4, π/4) ; +∞, θ∈[π/4,3π/4) .
In particular, given any smooth, odd functionψ: (−θ0, θ0)→R, with 0< θ0 ≤3π/4,
satisfying A(θ)< ψ(θ)< B(θ), the function
f =rexp
Z θ
0
ψ(σ)dσ
is a positive admissible speed function on the cone Γθ0 :={z∈R
2 :θ(z)∈(−θ 0, θ0)}.
Flows by concave speeds
Example 2.4 (Concave admissible flow speeds). The following symmetric functions define concave admissible flow speeds:
1. The power means Hr withr≤1 define positive concave admissible flow speeds.
2. The consecutive ratios of the elementary symmetric polynomials, Sk
Sk−1,k= 1, . . . , n
define positive concave admissible flow speeds (see, for example, Lieberman 1996, Chapter XV).
3. Concave combinations: Iffi : Γ→R,i= 1, . . . , N define concave admissible speeds,
and φ: ⊕N
i=1fi(Γ) ⊂ RN → R is a smooth (positive) concave, degree one homoge- neous function, then the function
f :=φ(f1, . . . , fN)
defines a (positive) concave admissible flow speed. In particular, (positive) linear combinations of (positive) concave admissible speeds are (positive) concave admissi- ble speeds and geometric means of positive admissible speeds are positive admissible speeds.
4. The roots of ratios of the elementary symmetric polynomials,
Sk
Sl
k1−l
, 0≤l < k≤ n, define positive concave admissible flow speeds.
Flows by convex speeds
Example 2.5 (Convex admissible flow speeds). The following symmetric functions define convex admissible flow speeds:
1. The power means: Hr for r ≥ 1 on the cone Γr := {z ∈ Rn : Hr(z) > 0, zir−1 >
0 for each i}.
2. Positive linear combinations of positive, convex admissible flow speeds define pos- itive, convex admissible flow speeds. For example, the functions of the form
f := P
rωrHr, ωr > 0, define positive, convex, admissible flow speeds on cones
containing Γ+. In particular, fε:=H1+√εnH2,ε∈(0,1), defines a positive, convex
admissible flow speed on the round cone Γε := {z ∈ Rn :H1(z) + √εnH2(z) > 0}.
We note that Γε contains the positive mean half-space.
3. Convex combinations: If fi : Γ→ R,i= 1, . . . , N define convex admissible speeds, and φ:⊕N
i=1fi(Γ) is a smooth (positive) convex, degree one homogeneous function,
then the function
f :=φ(f1, . . . , fN)
defines a (positive) convex admissible flow speed. For example, the function
fε(z1, . . . , zn) = Hr(z1 +εH, . . . , zn+εH), r ≥ 1 on the cone Γε := {z ∈ Rn :
zi+εH >0 for each i} defines a convex admissible speed.
4. Concave functions: If g : Γ → R is a concave admissible speed, then the function
f := H−εg on Γε := {z ∈ Γ : ˙gi < 1ε for each i, (H > εg)} defines a (positive)
convex admissible speed. Flows by inverse-concave speeds
Example 2.6 (Inverse-concave admissible flow speeds (cf. Andrews (2007) and Andrews, McCoy, and Zheng (2013))). The following symmetric functions define inverse-concave admissible flow speeds:
1. Convex admissible speeds f : Γ+→Rare inverse-concave (this follows from Lemma 2.11). In particular, the power means, Hr with r≥1 are inverse-concave.
2. Concave admissible speeds: If f : Γ+ → R is a concave admissible speed, then
f∗ : Γ+ → R is an inverse-concave admissible speed. For example, the harmonic mean H−1 = (H1)∗ is inverse-concave.
3. Iff : Γ+→Ris an inverse-concave admissible speed andr ∈[0,1], then the function
fr : Γ+→Rdefined by
fr(z1, . . . , zn) := (f(z1r, . . . , znr)) 1
r (2.24)
defines an inverse-concave admissible speed function (see Andrews 2007, Theorem 3.2).
4. If f : Γ+ → R is a concave admissible speed and r ∈ [−1,0], then the function
fr : Γ+→ Rdefined by (2.24) defines an inverse-concave admissible speed function (see Andrews 2007, Theorem 3.2).
5. The power means Hr withr∈[−1,1] are therefore concave, inverse-concave admis-
sible speeds.
6. The ratios of consecutive elementary symmetric polynomials, f := Sk
Sk−1, 0< k≤n,
are concave, inverse-concave admissible speeds, since f is concave and f∗ = SSn−k+1
n−k
is of the same type.
7. Inverse-concave combinations: If fi, i = 1, . . . , N are (concave) inverse-concave
admissible speeds, and φ : ΓN+ → R is a strictly monotone increasing, degree one homogeneous (concave) inverse-concave function, then the function
f :=φ(f1, . . . , fn)
is a (concave) inverse-concave admissible speed. In particular, positive linear com- binations and weighted geometric means of (concave) inverse-concave speeds are (concave) inverse-concave speeds.
8. The roots of ratios of the elementary symmetric polynomials,f :=Sk
Sl
k1−l
, 0≤l < k≤n, are concave, inverse-concave admissible speeds.
Flows which admit preserved cones
Example 2.7 (Admissible speeds whose flows admit preserved cones). The following sym- metric functions define admissible speeds which give rise to flows that admit preserved cones:
1. Surface flows by positive admissible speeds admit preserved cones (Corollary 4.15). 2. Flows by positive, convex admissible speeds f : Γ → R satisfying Γ+ ⊂ Γ admit
preserved cones (Corollary 4.19).
3. Flows by inverse-concave admissible speeds admit preserved cones (this follows, for example, from Theorem 6.1. See also Andrews (2007)).
4. Admissible flow speeds f : Γ→Rfor which Γf >0 ⊂Γ\ {0}, where Γf >0:={z∈Γ :
f(z)>0} preserve the cone Γf >0 (Proposition 4.5); for example, this holds for the
speedf :=H1−√1nH2, and many similar speeds which are admissible on Γ =Rn. 5. Concave admissible speeds f : Γ → R for which lim infλ→∂ΓHf1 > C preserve the
cone ΓC :={λ∈Γ :H1(λ)≤Cf(λ)} (Proposition 4.12).
6. Concave admissible speeds f : Γ→R such thatf = 0 on ∂Γ. This is a special case of the previous example. It holds, for example, for the speedsHr: Γ+→R,r ≤0.
In this section we will derive several results about the flow equation (CF) and its solutions. We begin by describing some invariance properties, and use these to construct some special solutions of the flow. Next, we introduce the linearized flow equation, and use the invari- ance properties of (CF) to construct some special solutions of the linearized equation. We then prove local existence of solutions of the initial value problem for (CF), which we do by reducing the flow equation to an equivalent scalar equation, and then appealing to a known existence result for (fully non-linear) scalar parabolic equations.
3.1
Invariance properties
We begin by deriving some invariance properties of the equation (CF), which allow us to generate new solutions from old.
3.1.1 Time translation
The simplest invariance property is invariance under time translation: Let X : M ×
(t1, t2)→Rn+1 be a solution of (CF). Then the familyXτ :M×(t1−τ, t2−τ)→Rn+1 defined byXτ(x, t) :=X(x, t+τ) also solves (CF), since∂tXτ(x, t) =∂tX(x, t+τ) and Wτ(x, t) =W(x, t+τ), whereWτ is the Weingarten map ofXτ.
3.1.2 Ambient isometries
Since (CF) is defined in terms of the induced geometry ofX, we expect that it should be invariant under isometries of the ambient space, and indeed this is the case, so long as the isometry is orientation preserving1: LetX :M ×I →Rn+1 be a solution of (CF)
and let Φ be an isometry of Rn+1. Then the familyXΦ : M ×I → Rn+1 defined by
e
X(x, t) = Φ(X(x, t)) is also a solution of (CF). This is because Φ is affine (and hence its second derivative vanishes) and the induced Weingarten map is invariant under ambient isometries: First note that
∂tXΦ(x, t) = Φ∗∂tX(x, t) =−F(x, t)Φ∗ν(x, t).
1Orientation reversing isometries leave the flow invariant ifFis given by an odd function of the principal
curvatures (See§3.1.6).
Next we compute, with respect to some local co¨ordinates,
∂iXΦ= Φ∗∂iX .
In particular, the induced metric and normal forXΦare given bygΦij =gij, andνΦ= Φ∗ν.
Now, since Φ is affine, we obtain
∂i∂jXΦ= Φ∗∂i∂jX.
It follows that the Weingarten map WΦ of XΦ satisfies WΦij = Wij, so that FΦ :=
F(WΦ)νΦ=F(W)Φ∗ν=FΦ∗ν as required.
3.1.3 Reparametrization
LetX :M×I →Rn+1 be a solution of (CF) and letφbe a diffeomorphism ofM. Then
the time-dependent immersionXφ : M ×I → Rn+1 defined byXφ(x, t) = X φ(x), t
satisfies
∂tXφ(x, t) =∂tX(φ(x), t) =−F(W(φ(x), t))ν(φ(x), t) =−F(Wφ(x, t))νφ(x, t),
where Wφ and νφ are, respectively, the Weingarten map and normal ofXφ. ThusXφ is
also a solution of (CF).
3.1.4 Time-dependent reparametrization
We observe that the previous calculation does not, in general, work if the reparametrization depends on time: LetX :M×I →Rn+1 be a solution of (CF) and letϕ:M×I →M be a time-dependent diffeomorphism (a smooth one parameter family of diffeomorphisms
ϕ(·, t)). Then the new time-dependent immersion Xϕ : M ×I → Rn+1 defined by Xϕ(x, t) :=X(ϕ(x, t), t) satisfies
∂tXϕ(x, t) =X∗(ϕ(x,t),t)∂tϕ(x, t)−F(ϕ(x, t), t)ν(ϕ(x, t), t).
So ∂tXϕ has an extra tangential term,X∗∂tϕ.
Thus, (CF) is not invariant under time-dependent diffeomorphisms of M; however, this calculation has a useful converse: Suppose that Y :M×I →Rn+1 satisfies
h∂tY , νi=−F .
Then, if we setX(x, t) := Y (ϕ(x, t), t) for some time-dependent diffeomorphism ϕ, we obtain
∂tX =Y∗∂tϕ+T−F ν ,
hypersurface. If we now letϕbe the solution of the ordinary differential equation ( Y∗∂tϕ=−T ϕ(·,0) = id, then we obtain ∂tX =−F ν .
Therefore, any solution of the equation h∂tY , νi = −F gives rise to a solution of (CF)
via a (unique) time-dependent reparametrization. 3.1.5 Space-time rescaling
Homogeneity of the speed implies a further useful invariance property: Observe that dilation of a hypersurface by a factor λ >0 rescales the Weingarten curvature, and, due to homogeneity, the speed, by a factorλ−1. This factor can be compensated by rescaling the time variable by a factor λ−2: LetX : M ×I → Rn+1 be a solution of (CF) and suppose λ >0. DefineXλ(x, t) :=λX(x, λ−2t). Then
∂tXλ(x, t) =λ−1∂tX(x, λ−2t)
= −λ−1F(W(x, λ−2t))ν(x, λ−2t) = −λ−1F(λWλ(x, t))νλ(x, t)
= −F(Wλ(x, t))νλ(x, t),
whereνλ and Wλ are the normal and corresponding Weingarten map ofXλ. 3.1.6 Orientation reversal
If the speed function is an odd function of the curvature, then the flow is also invariant under orientation reversals, since in that case
−F(Wν)ν =F(−W−ν)(−ν) =−F(W−ν)(−ν),
whereWν denotes the Weingarten map of ν and W−ν the Weingarten map of −ν.