Flow of real hypersurfaces by the trace of the Levi form

Gerhard Huisken  Wilhelm Klingenberg

1 Introduction.

Let F_{0} : N^{2n−1} → ℂ^{n}, n ≥ 2, be a smooth immersion of a real (2n − 1)-dimensional hypersurface in complex n-space. In this paper we study the Cauchy-Riemann analogue of mean curvature flow [?] in Riemannian geometry: We deform the initial hypersurface N_{0} = F_{0}(N^{2n−1}) in normal direction such that the speed at each point is given by the trace of the Levi form of the induced CR-structure on the hypersurface, with the defining one-form of norm one. For simplicity we assume in this paper that N^{2n−1} is closed, i.e. compact without boundary. Our initial value problem for the flow along the trace of the Levi form then looks for a smooth family of immersions F : N^{2n−1}×[0, T) → ℂ^{n} satisfying the system

(1.1)  \frac{d}{dt}F (p, t) = L(p, t) · X_{2n}(p, t)
F (p, 0) = F_{0}(p),

where p ∈ N^{2n−1}, t ∈ [0, T). Here X_{2n}(p, t) is the real unit normal of the hypersurface N_{t} = F (·, t)(N^{2n−1}) at F (p, t) in ℂ^{n} (equipped with the standard euclidean metric) and L(p, t) is the trace of the Levi form on N_{t} at F (p, t). On a closed hypersurface we choose X_{2n} to be the exterior unit normal and define the Levi form such that L is negative on the metric sphere in order to be consistent with the notation of [K].

This evolution equation is a weakly parabolic system exhibiting two different types of degeneracies: First, due to the invariance under tangential diffeomorphisms, the symbol is degenerate in tangential directions, a phenomenon which is well-known from mean curvature flow and can easily be dealt with by choosing a particular gauge. Secondly and more seriously, the trace of the Levi form L corresponds to a quasilinear second order differential operator ^{H}Δ on N^{2n−1} as in [K. 4.1], which acts like a Laplacian in complex directions HN := T N ∩ ^{ℂ^{n}}JT N of N but has a zero eigenvalue in its symbol in the direction of T N that is orthogonal to HN. After recalling the geometry of real hypersurfaces with their basic invariants from [?] in section 2, we prove shorttime existence of solutions of (1.1) in section 3. The proof is based on uniform a priori estimates for the curvature and its higher derivatives of solutions to approximating strictly parabolic systems. The necessary evolution equations for the curvature and its derivatives are computed with the help of commutator identities for the Kählerian shape operator established in [?]. In addition to local existence we show that the solutions remain smooth as long as both the curvature and its first derivatives remain bounded. This result differs from standard mean curvature flow and is due to the degenerate parabolic structure of the flow. In section 4 we exploit the parabolic nature of (1.1) in establishing a barrier principle for distinct solutions of the flow. We also show that embedded solutions stay embedded. In the case n = 2, where the Levi form reduces to a real scalar function, we prove a version of the strict maximum principle to show that smooth, closed weakly pseudoconvex surfaces instantaneously become strictly pseudoconvex under the evolution (1.1). This provides a canonical approximation of such weakly pseudoconvex surfaces by strictly pseudoconvex surfaces. We expect stronger regularity properties of the flow in the strictly pseudoconvex case, which will be examined in a future article. We refer to [?] for a weak solution of (1.1) in the level set formulation. Acknowledgement: We are grateful to Mrs. S. Schmidt for typing the manuscript. Remark: After completing our work, we became aware of the reference [?] and other work quoted therein that deals with the stationary problem of (1.1) in the case of a graph.

2 Notation and preliminaries.

Here we review the notation and results in [?] that will be used throughout this paper. For a real hypersurface F : N → ℂ^{n} let HN = T V ∩ ^{ℂ^{n}}JT N be the maximal complex tangent bundle in T N. We denote by T^{1,0} and T^{0,1} the eigenspaces for i and −i of ℂ⊗T ℂ^{n}. The Kählerian geometry of a hypersurface can be extracted from the principle bundle of unitary frames {e_{j}}_{1}^{n} of F^{∗}T^{1,0}ℂ^{n} that are adapted to T N ↪ F^{∗}T ℂ^{n} in the following way: span\{e_{α}\}_{1}^{n−1} = H^{1,0}N, span\{f_{α} := e_{α}, fᾱ := eᾱ, f_{n} := \frac{1}{2}(e_{n} + en̄)\} = ℂ ⊗ T N. We set X_{j} = \frac{1}{\sqrt{2}}(e_{j} + ej̄), X_{n+j} = \frac{i}{\sqrt{2}}(e_{j} − ej̄). It follows that X_{2n} is normal to T N. We have the following convention for ranges of indices: j, k, p, q ∈ {1, …, n}, α, β, γ ∈ {1, …, n − 1}, ξ, η ∈ \{1, …, n − 1, ¯1, …, \overline{n − 1}\}, a, b, c ∈ \{1, …, n, ¯1, …, \overline{n}\}. It follows for example that span{f_{α}} = H^{1,0}N, span{f_{ξ}} = ℂ ⊗ HN, span{f_{a}} = ℂ ⊗ T N. On the bundles HN ↪ T N ↪ F^{∗}T N over N we have the Levi-Civita connection ^{ℂ^{n}}∇ and its projections N∇, ^{H}∇ with connection forms ^{ℂ^{n}}ω, ^{N}ω, ^{H}ω and curvature forms ^{ℂ^{n}}Ω, ^{N}Ω, ^{H}Ω; see [?, 2.] for their structure equations and interrelations. In [?, 1.] Klingenberg defines a tensor ℓ on ℂ ⊗ T N that takes the role of a Kählerian shape operator of N → ℂ^{n}:

ℓ_{ja} := θ^{n}(^{ℂ^{n}}∇_{f_{a}}e_{j}), j = 1, …, n, a = 1, …, n − 1, ¯1, …, \overline{n − 1}, n.

Here, {θ^{j}} is the dual frame of {e_{j}}, {ℓ_{αβ̄}}_{1}^{n−1} is the Levi form of N → ℂ^{n}, which is hermitian on H^{1,0}N ⊗ H^{0,1}N, {ℓ_{αβ}} is symmetric on H^{1,0}N ⊗ H^{1,0}N, and {ℓ_{na}} is a real form on ℂ ⊗ T N. We will also denote by {ξ^{j}, ξ^{n+j}} the dual frame of {X_{j}, X_{n+j}} and {φ^{a}} the dual frame of {f_{a}}. The trace of the Levi form is given by

L := Σ_{1}^{n−1}ℓ_{αᾱ}.

It equals the negative trace of the restriction of the Riemannian second fundamental form to HN ↪ T N, see [K, 1.4]. In addition, the total norm |A|^{2} of the Riemannian second fundamental form is controlled by the total norm of ℓ_{ja}:

|A|^{2} ≤ 2|ℓ|^{2}, |ℓ|^{2} := Σ_{ja}|ℓ_{ja}|^{2}. (2.1)

We also set ℓ_{ab}^{2} = ℓ_{aγ̄}ℓ_{γb} and ℓ_{ab}^{3} = ℓ_{aγ̄}ℓ_{γδ̄}ℓ_{δb}. One can always choose a unitary frame {e_{α}} of H^{1,0}N so that ℓ_{αβ̄} = 0 for α ≠ β at a given point p ∈ N. We denote by ℓ_{jab}^{H} and ℓ_{jab}^{N} the components of the first covariant derivative of ℓ with respect to the connection on HN and T N. Finally we note that with the notation ^{H}Δ = ^{H}∇_{e_{ξ}}^{H}∇_{eξ̄} −^{H} ∇_{^{H}∇_{e_{ξ}} eξ̄} ≡ trace_{H} (^{H}∇^{2}), (1.1) has the form

\frac{d}{dt}F (p, t) = ^{H}ΔF (p, t),

which is analogous to the mean curvature flow in [H] given by

\frac{d}{dt}F (p, t) = ^{N}ΔF (p, t).

3 Short time existence.

The aim of this section is to establish two main existence results. First we prove short time existence of a solution of (??) with smooth initial data.

Theorem 3.1. Given a smooth regular immersion F_{0} : N^{2n−1} → ℂ^{n} of a closed hypersurface in complex Euclidean space, there exists a time interval [0, t_{0}), t_{0} > 0, with a smooth regular solution F : N^{2n−1} × [0, t_{0}) → ℂ^{n} of (??).

We can also show that the solution will continue to exist as long as the total curvature and the total gradient of the evolving hypersurface remains bounded. Notice that this result is in contrast to the mean curvature flow [H], where only the curvature needs to be bounded to establish longtime existence. The stronger condition here is due to the degeneracy in this flow which inhibits diffusion in the direction normal to HN. We expect a better regularity result in the strictly pseudoconvex case.

Theorem 3.2. There exists a maximal time interval [0, T), T > 0, admitting a smooth regular solution of (1.1). As t → T, the total curvature sup_{N_{t}} |ℓ|^{2}(t) or its total derivative

sup_{N_{t}} |∇ℓ|^{2}(t) becomes unbounded.

Remarks: i) The a priori estimates for the proof of Theorem 3.1 will also yield a lower bound for T in terms of the initial curvature and initial gradient of curvature, see Corollary 3.6. ii) We will show in section 4 that for compact initial data T is bounded above in terms of the radius R_{0} of the smallest sphere containing N_{0} : T ≤ R_{0}^{2}/4(n − 1). iii) Since equation (??) is more degenerate than mean curvature flow, we cannot expect a general smoothing behaviour like in that case and Theorem 3.1 depends heavily on the smoothness of the initial data. However, it will become apparent in the proof that for intitial data in C^{k−1,1}, k ≥ 3, a solution exists in C^{k−1,1}. Whether some smoothing result is true, e.g. for strictly pseudoconvex initial data, is an interesting open problem. iv) Theorems 3.1 and 3.2 can be extended to hypersurfaces in arbitrary Kähler manifolds, since the corresponding evolution equations for ℓ_{ja} differ only by terms involving the given curvature tensor of the ambient manifold and are of lower order in the estimates derived here.

To prove the two theorems we first consider smooth regular solutions of the evolution equation for a regular F : N × [0, T) → ℂ^{n}:

\frac{d}{dt}F = KX_{2n} on N^{2n−1} × [0, T), (3.1)

with an arbitrary speed function K, and establish how the curvature changes under this flow. Given an adapted frame {e_{k}} of F^{∗}(·, 0)T^{1,0}ℂ^{n}, we need to define its time evolution in F^{∗}(·, t)T^{1,0}ℂ^{n} over N_{t}. For that we extend F to a map F̃: N × [0, T) → ℂ^{n} × ℝ, F̃(p, t) = (F (p, t), t). Then the bundle F̃^{∗}T (ℂ^{n} × ℝ) = F^{∗}T ℂ^{n} ⊕ ℝ over N × [0, T) can be equipped with the product metric in each fibre and the Levi-Civita connection ^{ℂ^{n}×ℝ}∇. In addition, {e_{k}} is naturally a frame of F̃^{∗}(·, 0)(T^{1,0}ℂ^{n} ⊕ 0) = F^{∗}(·, 0)T^{1,0}ℂ^{n} ⊕ 0 over N_{0} × 0. Extend ^{ℂ^{n}×ℝ}∇ trivially to T F̃^{∗}(·, 0)(T ℂ^{n} ⊕ ℝ) = F^{∗}(T ℂ^{n} ⊕ ℝ) ⊕ ℂ^{n} ⊕ ℝ and set

Z := F̃^{∗}F̃_{∗}(∂/∂t).

Then Z = (KX_{2n}, 1) is transverse to N × t for every t. Setting K_{a}:= f_{a} · K, the following system of ODEs then uniquely determines an evolution for {e_{k}}:

∇_{Z}e_{α} = i\sqrt{2}K_{α}e_{n}, (3.2)

∇_{Z}e_{n} = i\sqrt{2}Kj̄e_{j}. (3.3)

This is well posed since for every t, the vector Z is transverse to N × t in T F̃^{∗}(T ℂ^{n} ⊕ 0). In addition by the Kähler property, this evolution gives a frame of F̃^{∗}(·, t)(T^{1,0}ℂ^{n} ⊕ 0). We finally extend the frame by parallel transport along the vectorfield X_{2n} to T F̃^{∗}(T^{1,0}ℂ^{n} ⊕ 0). This implies that ∇_{Z}e_{j} = ∇_{\frac{∂}{∂t}} e_{j} and we will often simply write \frac{∂}{∂t} to denote such a derivative. We claim that the above evolution implies

(3.4)  ξ^{2n}(L_{Z}f_{a}) = 0,
(3.5)  θ^{β}(^{ℂ^{n}×ℝ}∇_{Z}e_{α}) = 0.

Thus our choice of evolution for the frame {e_{j}} ensures that the frame remains adapted to HN_{t} ⊂ T N_{t}, without rotating the holomorphic directions. To prove the claim, note that (3.3) implies ∇_{∂/∂t}X_{2n} = −K_{ξ}fξ̄ − 2K_{n}f_{n}, and therefore φ^{n}(∇_{∂/∂t}X_{2n}) = −2K_{n}, φξ̄(∇_{∂/∂t}X_{2n}) = −K_{ξ}, which gives ξ^{2n}(∇_{∂/∂t}f_{a}) = K_{a} and implies (3.4). Also, (3.2) clearly implies (3.5). This proves the claim. In addition, we infer the identities

(3.6)  L_{∂/∂t}f_{α} = \frac{1}{2}Kℓ_{αξ̄}f_{ξ} + (Kℓ_{αn} + i\sqrt{2}K_{α})f_{n},
(3.7)  L_{∂/∂t}f_{n} = \frac{1}{2}Kℓ_{nξ̄}f_{ξ} + \frac{i}{\sqrt{2}}(Kγ̄f_{γ} − K_{γ}fγ̄) + Kℓ_{nn}f_{n}.

Proposition 3.3 For the evolution (??) of hypersurfaces, we have the following evolution of ℓ with respect to the frame constructed above.

\frac{∂}{∂t}ℓ_{αξ} = 2(^{H}∇^{2}K)(f_{α}, f_{ξ}) + i\sqrt{2}(ℓ_{nξ}K_{α} ± ℓ_{nα}K_{ξ} − ℓ_{αξ}K_{n})

+\frac{1}{2}Kℓ_{αη̄}ℓ_{ηξ} + Kℓ_{αn}ℓ_{nξ}

\frac{∂}{∂t}ℓ_{nα} = 2 (^{H}∇^{2}K)(f_{n}, f_{α}) + i\sqrt{2}(ℓ_{αγ}Kγ̄ − ℓ_{αγ̄}K_{γ} + ℓ_{nn}K_{α} − ℓ_{αn}K_{n})

+\frac{1}{2}Kℓ_{αη̄}ℓ_{ηn} + Kℓ_{αn}ℓ_{nn}

\frac{∂}{∂t}ℓ_{nn} = 2 (^{H}∇^{2}K)(f_{n}, f_{n}) + \frac{3i}{\sqrt{2}}(ℓ_{nγ}Kγ̄ − ℓ_{nγ̄}K_{γ})

+\frac{1}{2}Kℓ_{nη̄}ℓ_{ηn} + K(ℓ_{nn})^{2}.

In the first equation, the upper sign holds if ξ ∈ {1, …, n − 1} and the lower sign holds otherwise. Remark: It is easy to see that the evolution equations above only depend on the construction of the frame on N × [0, T), not on the parallel extension of the frame along X_{2n} in the ambient space. For a different extension of the frame to the ambient space the time derivative is then replaced by differentiation in direction Z. Proof: By the product rule,

\frac{∂}{∂t}θ^{n}(∇_{f_{a}}e_{j}) = (∇_{∂/∂t}θ^{n})(∇_{f_{a}}e_{j}) + θ^{n}(∇_{[∂/∂t,f_{a}]}e_{j}) + θ^{n}(∇_{f_{a}}∇_{∂/∂t}f_{a}).

Using the relations (3.2)–(3.7) and the identities f_{a}·K_{α} = (^{H}∇^{2}K)(f_{α}, f_{a})K +K_{δ}ω_{α}^{δ}(f_{a}), f_{a} · K_{n} = (^{H}∇^{2}K)(f_{n}, f_{a}) , we compute the above expression for various (a, j).

\frac{∂}{∂t}θ^{n}(∇_{f_{ξ}}e_{α}) = −i\sqrt{2}K_{j}θ^{j}(∇_{f_{ξ}}e_{α}) + \frac{1}{2}Kℓ_{ξη̄}θ^{n}(∇_{f_{η}}e_{α})
+(Kℓ_{nξ} ± i\sqrt{2}K_{ξ})θ^{n}(∇_{f_{n}}e_{α})
+i\sqrt{2}K_{α}θ^{n}(∇_{f_{ξ}}e_{n}) + i\sqrt{2} (^{H}∇^{2}K)(f_{α}, f_{ξ})
+i\sqrt{2}K_{δ}ω_{α}^{δ}(f_{ξ})

= −i\sqrt{2}K_{n} \frac{i}{\sqrt{2}}ℓ_{αξ} + \frac{i}{2^{3/2}} Kℓ_{ξη̄}ℓ_{ηα}

+\frac{i}{\sqrt{2}}Kℓ_{nξ}ℓ_{nα} ∓ ℓ_{nα}K_{ξ} − ℓ_{nξ}K_{α}

+i\sqrt{2} (^{H}∇^{2}K)(f_{α}, f_{ξ})

\frac{∂}{∂t}θ^{n}(∇_{f_{α}}e_{n}) = −i\sqrt{2}K_{j}θ^{j}(∇_{f_{α}}e_{n}) + \frac{1}{2}Kℓ_{αξ̄}θ^{n}(∇_{f_{ξ}}e_{n})

+(Kℓ_{αn} + i\sqrt{2}K_{α})θ^{n}(∇_{f_{n}}e_{n}) + i\sqrt{2}Kj̄θ^{n}(∇_{f_{α}}e_{j})

+i\sqrt{2} (^{H}∇^{2}K)(f_{n}, f_{α})

= K_{γ}ℓ_{αγ̄} + \frac{i}{2^{3/2}} Kℓ_{αξ̄}ℓ_{ξn} + \frac{i}{\sqrt{2}}Kℓ_{αn}ℓ_{nn}

−K_{α}ℓ_{nn} − Kγ̄ℓ_{αγ} + i\sqrt{2} (^{H}∇^{2}K)(f_{n}, f_{α})

\frac{∂}{∂t}θ^{n}(∇_{f_{n}}e_{n}) = −i\sqrt{2}K_{j}θ^{j}(∇_{f_{n}}e_{n}) + \frac{1}{2}Kℓ_{nη̄}θ^{n}(∇_{f_{η}}e_{n})

+\frac{i}{\sqrt{2}}(Kγ̄θ^{n}(∇_{f_{γ}}e_{n}) − K_{γ}θ^{n}(∇_{fγ̄}e_{n}))

+Kℓ_{nn}θ^{n}(∇_{f_{n}}e_{n}) + i\sqrt{2}(^{H}∇^{2}K)(f_{n}, f_{n}) + i\sqrt{2}Kj̄θ^{n}(∇_{f_{n}}e_{j})

= K_{γ}ℓ_{nγ̄} + \frac{i}{2^{3/2}} Kℓ_{nη̄}ℓ_{ηn} − \frac{1}{2}Kγ̄ℓ_{γn}

+\frac{1}{2}K_{γ}ℓ_{nγ̄} + \frac{i}{\sqrt{2}}K(ℓ_{nn})^{2} + i\sqrt{2} (^{H}∇^{2}K)(f_{n}, f_{n}) − Kγ̄ℓ_{nγ}.

These easily imply the claims. QED

We will apply Theorem 3.1 with K = L + εL^{N}, where −L^{N} is the Riemannian mean curvature of the hypersurface. The parabolic nature of the equations in Theorem 3.1 will become apparent by combining them with the commutator equations established in [?, 4.2].

Proposition 3.4 Let □_{ε} = \frac{∂}{∂t} − (^{H}Δ + ε ^{N}Δ). Then

□_{ε}ℓ_{αβ̄} = i\sqrt{2}[ℓ_{nγ̄}ℓ_{αβ̄γ}^{H} − ℓ_{nγ}ℓ_{αβ̄γ̄}^{H} + 2(ℓ_{γβ̄}ℓ_{αγ̄n}^{H} − ℓ_{αγ̄}ℓ_{γβ̄n}^{H})]

+\frac{1}{2}ℓ_{αβ̄}(ℓ_{γγ̄}^{2} + ℓ_{γ̄γ}^{2} + 2ℓ_{nn}^{2}) + \frac{1}{2}(ℓ_{αγ}^{2}ℓγ̄β̄ + ℓ_{αγ}ℓ_{γ̄β̄}^{2}) − ℓ_{β̄α}^{3}

+ε[i\sqrt{2}(ℓ_{nβ̄}L_{α}^{N} − ℓ_{nα}LN ; β̄) + |ℓ|^{2}ℓ_{αβ̄}]

□_{ε}ℓ_{αβ} = i\sqrt{2}[ℓ_{nγ̄}ℓ_{αβγ}^{H} − ℓ_{nγ}ℓ_{αβγ̄}^{H} − 2ℓ_{αγ̄}ℓ_{γβn}^{H} + ℓ_{βγ̄}^{2}ℓ_{γαn}^{H})]

+\frac{1}{2}ℓ_{αβ}(ℓ_{γγ̄}^{2} + ℓ_{γ̄γ}^{2} + 2ℓ_{nn}^{2}) + 2ℓ_{nn}(ℓ_{αβ}^{2} + ℓ_{βα}^{2})

−4(ℓ_{αn}^{2}ℓ_{nβ} + ℓ_{αn}ℓ_{βn}^{2}) + \frac{1}{2}(ℓ_{αγ}ℓ_{γ̄β}^{2} + ℓ_{γ̄α}^{2}ℓ_{γβ}) − \frac{1}{2}(ℓ_{αβ}^{3} + ℓ_{βα}^{3})

+ε[i\sqrt{2}(ℓ_{nβ}L_{α}^{N} + ℓ_{nα}L_{β}^{N}) + |ℓ|^{2}ℓ_{αβ}]

□_{ε}ℓ_{nα} = i\sqrt{2}[ − 2ℓ_{αγ̄}ℓ_{γnn}^{H} + \frac{1}{2}(ℓ_{γδ}ℓ_{γ̄δ̄α}^{H} − ℓγ̄δ̄ℓ_{γδα}^{H}) − ℓ_{αn}L_{n}]

+ℓ_{nα}(ℓ_{γγ̄}^{2} + ℓ_{γ̄γ}^{2}) − 2(ℓ_{αγ̄}ℓ_{nγ}^{2} + ℓ_{nα}^{3})

+\frac{1}{2}(ℓ_{nγ}ℓ_{γ̄α}^{2} + ℓ_{αn}^{3})

+ε[i\sqrt{2}(ℓ_{αγ}LN ; γ̄ − ℓ_{αγ̄}L_{γ}^{N} + ℓ_{nn}L_{α}^{N}) + |ℓ|^{2}ℓ_{αn}]

□_{ε}ℓ_{nn} = i\sqrt{2}(ℓ_{nγ}ℓ_{nnγ̄} − ℓ_{nγ̄}ℓ_{nnγ} + ℓ_{γδ}ℓ_{γ̄δ̄n} − ℓγ̄δ̄ℓ_{γδn})

+\frac{3}{2}ℓ_{nn}(ℓ_{γγ̄}^{2} + ℓ_{γ̄γ}^{2} + \frac{3}{2}ℓ_{nn}^{2}) − 6ℓ_{nn}^{3} + \frac{1}{2}(ℓ_{γδ̄}ℓ_{γ̄δ}^{2} − ℓ_{γγ̄}^{3})

+ε[\frac{3i}{\sqrt{2}}(ℓ_{nγ}LN ; γ̄ − ℓ_{nγ̄}L_{γ}^{N} + \frac{2}{3}ℓ_{nn}L_{n}^{N}) + |ℓ|^{2}ℓ_{nn}].

Proof: Note that by definition of^{N}∇ and^{H}∇, as explained in section 2, (^{N}∇^{2}L^{N})(f_{j}, f_{a}) + L_{n}^{N} ^{N}ω_{j}^{n}(f_{a}) = (^{H}∇^{2}L^{N})(f_{j}, f_{a}) . By [?, 2.1] we have ^{N}ω_{j}^{n}(f_{a}) = ω_{j}^{n}(f_{a}) such that in view of the commutator identities from [?, 4.0, 4.2] for ^{N}Δ, ^{H}Δ we may proceed as follows with K = L + εL^{N}. In this computation we use Proposition 3.1 and the definition of ℓ_{(ab)}^{2} given in [K, 4.0].

□_{ε}ℓ_{αβ̄} = (\frac{∂}{∂t}ℓ_{αβ̄} − 2 (^{H}∇^{2}K)(f_{α}, fβ̄)) + (2 (^{H}∇^{2}L)(f_{α}, fβ̄) − ^{H}Δℓ_{αβ̄})

+ε(2 (^{N}∇^{2}L^{N})(f_{α}, fβ̄) + 2L_{n}^{N}ω_{α}^{n}(fβ̄) − ^{N}Δℓ_{αβ̄})

= i\sqrt{2}(ℓ_{nβ̄}K_{α} − ℓ_{nα}Kβ̄ − ℓ_{αβ̄}K_{n}) + \frac{1}{2}Kℓ_{αη̄}ℓ_{ηβ̄}

+Kℓ_{αn}ℓ_{nβ̄} + i\sqrt{2}[ℓ_{nγ̄}ℓ_{αβ̄γ}^{H} − ℓ_{nγ}ℓ_{αβ̄γ̄}^{H}

+2(ℓ_{γβ̄}ℓ_{αγ̄n}^{H} − ℓ_{αγ̄}ℓ_{γβ̄n}^{H}) + ℓ_{nα}Lβ̄ − ℓ_{nβ̄}L_{α} + ℓ_{αβ̄}L_{n}]

+\frac{1}{2}ℓ_{αβ̄}(ℓ_{γγ̄}^{2} + ℓ_{γ̄γ}^{2} + 2ℓ_{nn}^{2}) − \frac{1}{2}L(ℓ_{αβ̄}^{2} + ℓ_{β̄α}^{2} + 2ℓ_{nα}ℓ_{nβ̄})

+\frac{1}{2}(ℓ_{αγ}^{2}ℓγ̄β̄ + ℓ_{αγ}ℓ_{γ̄β̄}^{2}) − ℓ_{β̄α}^{3}

+ε(|ℓ|^{2}ℓ_{αβ̄} − L^{N}ℓ_{(αβ̄)}^{2} + i\sqrt{2}ℓ_{αβ̄} − L_{n}^{N})

□_{ε}ℓ_{αβ} = (\frac{∂}{∂t}ℓ_{αβ} − 2 (^{H}∇^{2}K)(f_{α}, f_{β})) + (2 (^{H}∇^{2}L)(f_{α}, f_{β}) − ^{H}Δℓ_{αβ})

+ε(2 (^{N}∇^{2}L^{N})(f_{α}, f_{β}) + 2L_{n}^{N} ^{N}ω_{α}^{n}(f_{β}) − ^{N}Δℓ_{αβ})

= i\sqrt{2}(ℓ_{nβ}K_{α} + ℓ_{nα}K_{β} − ℓ_{αβ}K_{n}) + \frac{1}{2}Kℓ_{αη̄}ℓ_{ηβ}

+Kℓ_{αn}ℓ_{βn} + i\sqrt{2}[ℓ_{nγ̄}ℓ_{αβγ}^{H} − ℓ_{nγ}ℓ_{αβγ̄}^{H}

−2(ℓ_{αγ̄}ℓ_{γβn}^{H} + ℓ_{βγ̄}ℓ_{γαn}^{H}) − ℓ_{nα}L_{β} − ℓ_{nβ}L_{α} + ℓ_{αβ}L_{n}]

+\frac{1}{2}ℓ_{αβ}(ℓ_{γγ̄}^{2} + ℓ_{γ̄γ}^{2} + 2ℓ_{nn}^{2})

−\frac{1}{2}L(ℓ_{αβ}^{2} + ℓ_{βα}^{2} + 2ℓ_{αn}ℓ_{βn}) + 2ℓ_{nn}(ℓ_{αβ}^{2} + ℓ_{βα}^{2})

−4(ℓ_{αn}^{2}ℓ_{nβ} + ℓ_{αn}ℓ_{βn}^{2}) + \frac{1}{2}(ℓ_{αγ}ℓ_{γ̄β}^{2} + ℓ_{γ̄α}^{2}ℓ_{γβ}) − \frac{1}{2}(ℓ_{αβ}^{3} + ℓ_{βα}^{3})

+ε(|ℓ|^{2}ℓ_{αβ} − L^{N}ℓ_{(αβ)}^{2} + i\sqrt{2}ℓ_{αβ}L_{n}^{N})

□_{ε}ℓ_{nα} = (\frac{∂}{∂t}ℓ_{nα} − 2 (^{H}∇^{2}K)(f_{n}, f_{α})) + (2 (^{H}∇^{2}L)(f_{n}, f_{α}) − ^{H}Δℓ_{nα})

+ε(2 (^{N}∇^{2}L^{N})(f_{n}, f_{α}) + 2L_{n}^{N} ^{N}ω_{n}^{n}(f_{α}) − ^{N}Δℓ_{nα})

= i\sqrt{2}(ℓ_{αγ}Kγ̄ − ℓ_{αγ̄}K_{γ} + ℓ_{nn}K_{α} − ℓ_{αn}K_{n})

+\frac{1}{2}Kℓ_{αη̄}ℓ_{ηn} + Kℓ_{αn}ℓ_{nn}

+i\sqrt{2}[ − 2ℓ_{αγ̄}ℓ_{nγn}^{H} − 2ℓ_{nn}L_{α} − ℓ_{αγ}Lγ̄

+ℓ_{αγ̄}L_{γ} + \frac{1}{2}(ℓ_{γδ}ℓ_{δ̄γ̄α} − ℓγ̄δ̄ℓ_{δγα})]

+ℓ_{nα}(ℓ_{γγ̄}^{2} + ℓ_{γ̄γ}^{2}) − \frac{1}{2}L(ℓ_{nα}^{2} + ℓ_{αn}^{2} + 2ℓ_{nα}ℓ_{nn})

−2(ℓ_{αγ̄}ℓ_{nγ}^{2} + ℓ_{nα}^{3}) + \frac{1}{2}(ℓ_{nγ}ℓ_{γ̄α}^{2} + ℓ_{αn}^{3})

+ε(|ℓ|^{2}ℓ_{αn} − L^{N}ℓ_{(αn)}^{2})

□_{ε}ℓ_{nn} = (\frac{∂}{∂t}ℓ_{nn} − 2(^{H}∇^{2}K)(f_{n}, f_{n})) + (2 (^{H}∇^{2}L)(f_{n}, f_{n}) − ^{H}Δℓ_{nn})

+ε(2 (^{N}∇^{2}L^{N})(f_{n}, f_{n}) + 2L_{n}^{N} ^{N}ω_{n}^{n}(f_{n}) − ^{N}Δℓ_{nn})

= \frac{3i}{\sqrt{2}}(ℓ_{nγ}Kγ̄ − ℓ_{nγ̄}K_{γ}) + \frac{1}{2}Kℓ_{nη̄}ℓ_{ηn} + K(ℓ_{nn})^{2}

+i\sqrt{2}[ℓ_{nγ}ℓ_{nnγ̄} − ℓ_{nγ̄}ℓ_{nnγ} + \frac{3}{2}(ℓ_{nγ̄}L_{γ} − ℓ_{nγ}Lγ̄)

+ℓ_{γδ}ℓ_{γ̄δ̄n} − ℓγ̄δ̄ℓ_{γδn}]

+\frac{3}{2}ℓ_{nn}(ℓ_{γγ̄}^{2} + ℓ_{γ̄γ}^{2} + \frac{2}{3}ℓ_{nn}^{2}) − L(ℓ_{nn}^{2} + (ℓ_{nn})^{2})

−6ℓ_{nn}^{3} + \frac{1}{2}(ℓ_{γδ̄}ℓ_{γ̄δ}^{2} − ℓ_{γγ̄}^{3})

+ε(|ℓ|^{2}ℓ_{nn} − L^{N}ℓ_{(nn)}^{2}).

Further simplification immediately leads to the claim. QED

The above Proposition implies the following estimate.

Proposition 3.5. Let 0 ≤ ε ≤ 1. Then there is a constant c_{0} depending only on n such that

□_{ε}|ℓ|^{2} ≤ −2|^{H}∇_{H}ℓ|^{2} + c_{0}|ℓ|^{4}.

Proof. For two tensors S, T we denote by S ∗T any tensor whose entries are homogeneous polynomials of degree two and are linear in both S and T. We set ∇ = ^{N}∇, then we denote by ∇_{H} the covariant derivative taken in direction HN. Similarly for the connection H∇, we have^{H}∇_{H} standing for the covariant derivative w.r.t. ^{H}∇, taken in direction HN. With this notation, we have (∇ − ^{H}∇)T = ℓ ∗ T, and Proposition 3.4 gives

□_{ε}ℓ_{pa} = ℓ ∗ ^{H}∇_{H}ℓ + ℓ ∗ ℓ ∗ ℓ + ε(ℓ ∗ ∇ℓ + ℓ ∗ ℓ ∗ ℓ).

Here we also used the Codazzi-type commutator relation [?, 2.4] to replace ^{H}∇L by H∇_{H}ℓ + ℓ ∗ ℓ. Therefore we have for any pair of indices (pa)

\frac{∂}{∂t}|ℓ_{pa}|^{2} = (^{H}Δ + ε^{N}Δ)|ℓ_{pa}|^{2} − 2| ^{H}∇_{H}ℓ_{pa}|^{2}
+ℓ ∗ ℓ ∗ ^{H}∇_{H}ℓ + ℓ ∗ ℓ ∗ ℓ ∗ ℓ
+ε(−2|∇ℓ_{pa}|^{2} + ℓ ∗ ℓ ∗ ∇ℓ + ℓ ∗ ℓ ∗ ℓ ∗ ℓ).

Now using Cauchy-Schwarz, this completes the proof. QED

Corollary 3.6 If at t = 0 we have a curvature bound of the form max_{N_{0}} |ℓ|^{2} ≤ B, then

max_{N_{t}} |ℓ|^{2} ≤ \frac{B}{1−c_{0}Bt}. In particular, the curvature will not blow up for t ∈ [0, c_{0}^{−1}B^{−1}).

Proof. Since □_{ε} is a (weakly) parabolic operator for 0 ≤ ε ≤ 1, we can apply the parabolic maximum principle to compare with the solution of the ODE ġ = c_{0}g^{2} with initial value B, which is given by g(t) = \frac{B}{1−c_{0}Bt}. QED

We will also need to control higher order derivatives and set |∇^{m}ℓ|^{2} = ∑_{p,a} |∇^{m}ℓ_{pa}|^{2}.

Proposition 3.7. Let 0 ≤ ε ≤ 1. (i) There are constants c_{1} and c_{2} depending only on n such that the first derivatives of the curvature satisfy the estimate

□_{ε}|∇ℓ|^{2} ≤ −|^{H}∇_{H}∇ℓ|^{2} + c_{1}|∇ℓ|^{3} + c_{2}|ℓ|^{2}|∇ℓ|^{2}.

(ii) For each m ≥ 2 there are constants c(n, m) depending only on n, m and max_{N×[0,T)} |∇^{i}ℓ|

for 0 ≤ i ≤ m − 1 such that

□_{ε}|∇^{m}ℓ|^{2} ≤ −|^{H}∇_{H}∇^{m}ℓ|^{2} + c(n, m)(1 + |∇^{m}ℓ|^{2}).

Proof. Since the good Bochner-type term on the RHS will look like −|^{H}∇_{H}∇^{m−1}ℓ|^{2}, which doesn’t dominate the total m-th derivative of ℓ, we need to keep track of the horizontal derivatives in the subsequent computations. We compute

(∇_{∂/∂t}∇_{f_{a}} − ∇_{f_{a}}∇_{∂/∂t} − ∇_{[∂/∂t,f_{a}]})f_{b} = (∇_{∂/∂t}θ^{c})(∇_{f_{a}}f_{b})f_{c}.

By (3.2),(3.3),(3.6) and (3.7), the coefficients of the RHS are polynomials of type ℓ ∗ ∇ℓ. By (3.6), (3.7), the third term on the LHS is of type (ℓ ∗ ℓ + ∇ℓ) ∗ ∇f_{b}. (Note that the term ∇ℓ does not occur in the mean curvature flow, where the frame is unrestrained in tangential directions. This additional term in the Levi flow causes the cubic term in the equation for the gradient above.) We conclude, for a tensor T,

(∇_{∂/∂t}∇_{f_{a}} − ∇_{f_{a}}∇_{∂/∂t})T = ℓ ∗ ∇ℓ ∗ T + (ℓ ∗ ℓ + ∇ℓ) ∗ ∇T,

such that in particular

(∇_{∂/∂t}∇^{m} − ∇^{m}∇_{∂/∂t})ℓ_{pa} = ∑_{r=0}^{m−1} ∇^{r}(∇_{∂/∂t}∇ − ∇∇_{∂/∂t})∇^{m−r−1}ℓ

= ∑_{r=0}^{m−1} ∇^{r}[ℓ ∗ ∇ℓ ∗ ∇^{m−r−1}ℓ

+ℓ ∗ ℓ ∗ ∇^{m−r}ℓ + ∇ℓ ∗ ∇^{m−r}ℓ]

= ∑_{i+j+k=m} ∇^{i}ℓ ∗ ∇^{j}ℓ ∗ ∇^{k}ℓ

+ ∑_{{i+j=m+1 \atop i,j≥1}} ∇^{i}ℓ ∗ ∇^{j}ℓ.

We also need commutator relations on ℂ ⊗ T N. By the Gauss-type equation in [?, 2.1], we have ^{N}Ω = ℓ ∗ ℓ. Also, when expressed as a linear combination of {f_{c}}, the vector [f_{a}, f_{b}] has coefficients of type ℓ. This gives (∇_{f_{a}}∇_{f_{b}} − ∇_{f_{b}}∇_{f_{a}})T = ℓ ∗ ∇T + ℓ ∗ ℓ ∗ T. In addition, since (∇ − ^{H}∇)T = ℓ ∗ T, we have

HΔT = (∇_{H} + ^{H}∇_{H} − ∇_{H})^{2}T
= ∇_{H}^{2}T + ∇_{H}ℓ ∗ T + ℓ ∗ ∇_{H}T + ℓ ∗ ℓ ∗ T.

Therefore we compute the commutator of the horizontal Laplacian

(∇^{m} ^{H}Δ − ^{H}Δ∇^{m})ℓ_{pa} = (∇^{m}∇_{H}^{2} − ∇_{H}^{2}∇^{m})ℓ + ∇^{m}(ℓ ∗ ∇_{H}ℓ + ℓ ∗ ℓ ∗ ℓ)
= ∑_{r=0}^{m−1} ∇^{r}(∇_{H}∇ − ∇∇_{H})∇^{m−r−1}∇_{H}ℓ
+∇^{m}(ℓ ∗ ∇_{H}ℓ + ℓ ∗ ℓ ∗ ℓ)

= ∑_{r=0}^{m−1} ∇^{r}(ℓ ∗ ∇^{m−r}∇_{H}ℓ + ℓ ∗ ℓ ∗ ∇^{m−r−1}∇_{H}ℓ)

+∇^{m}(ℓ ∗ ∇_{H}ℓ + ℓ ∗ ℓ ∗ ℓ)

= ∑_{i+j=m} ∇^{i}ℓ ∗ ∇^{j}∇_{H}ℓ + ∑_{i+j+k=m} ∇^{i}ℓ ∗ ∇^{j}ℓ ∗ ∇^{k}ℓ
= ∑_{i+j=m} ∇^{i}ℓ ∗^{H} ∇_{H}∇^{j}ℓ + ∑_{i+j+k=m} ∇^{i}ℓ ∗ ∇^{j}ℓ ∗ ∇^{k}ℓ.

Using then

(∇^{m} ^{N}∇_{f_{a}}^{N}∇_{f_{a}} − ^{N}∇_{f_{a}}^{N}∇_{f_{a}}∇^{m})ℓ_{pa} = ∑_{r=0}^{m−1} ∇^{r}(∇_{fa}∇ − ∇∇_{fa})∇^{m−r}ℓ

= ∑_{r=0}^{m−1} ∇^{r}(ℓ ∗ ∇^{m−r+1}ℓ + ℓ ∗ ℓ × ∇^{m−r}ℓ),

we conclude for the Riemannian Laplacian

(∇^{m} ^{N}Δ − ^{N}Δ ∇^{m})ℓ = ∑_{i+j=m+1} ∇^{i}ℓ ∗ ∇^{j}ℓ + ∑_{i+j+k=m} ∇^{i}ℓ ∗ ∇^{j}ℓ ∗ ∇^{k}ℓ.

Now, by Proposition 3.4 and the Codazzi equations established in [K,2.4], we may write the evolution equations of the curvature as

\frac{∂}{∂t}ℓ_{pa} = (^{H}Δ + ε ^{N}Δ)ℓ_{pa} + ℓ ∗ ^{H}∇_{H}ℓ + ℓ ∗ ℓ ∗ ℓ.

Combining these identities we obtain for the higher derivatives of the curvature

\frac{∂}{∂t}|∇^{m}ℓ_{pa}|^{2} = [(∇_{∂/∂t}∇^{m} − ∇^{m}∇_{∂/∂t})ℓ_{pa}

+∇^{m}(^{H}Δℓ_{pa} + ℓ ∗ ^{H}∇_{H}ℓ + ℓ ∗ ℓ ∗ ℓ)

+ε∇^{m} ^{N}Δℓ_{pa}]∇^{m}ℓ\overline{pa} + complex conjugate

= (^{H}Δ + ε ^{N}Δ)|∇^{m}ℓ_{pa}|^{2} − 2| ^{H}∇_{H}∇^{m}ℓ_{pa}|^{2}

−2ε|∇^{m+1}ℓ_{pa}|^{2} +(∇_{∂/∂t}∇^{m} − ∇^{m}∇_{∂/∂t})ℓ ∗ ∇^{m}ℓ +(∇^{m} ^{H}Δ − ^{H}Δ∇^{m})ℓ ∗ ∇^{m}ℓ +∇^{m}(ℓ ∗ ^{H}∇_{H}ℓ + ℓ ∗ ℓ ∗ ℓ) ∗ ∇^{m}ℓ +ε(∇^{m} ^{N}Δ − ^{N}Δ∇^{m})ℓ ∗ ∇^{m}ℓ.

Hence we finally conclude, setting |∇^{m}ℓ|^{2} := ∑_{p,a} |∇^{m}ℓ_{pa}|^{2},

□_{ε}|∇^{m}ℓ|^{2} = −2|^{H}∇_{H}ℓ|^{2} + ∑_{i+j=m} ∇^{i}ℓ ∗ ^{H}∇_{H}∇^{j}ℓ ∗ ∇^{m}ℓ

+ ∑_{{i+j=m+1 \atop i,j≥1}} ∇^{i}ℓ ∗ ∇^{j}ℓ ∗ ∇^{m}ℓ

+ ∑_{i+j+k=m} ∇^{i}ℓ ∗ ∇^{j}ℓ ∗ ∇^{k}ℓ ∗ ∇^{m}ℓ

+ε(−2|∇^{m+1}ℓ|^{2} + ∑_{i+j=m+1} ∇^{i}ℓ ∗ ∇^{j}ℓ ∗ ∇^{m}ℓ

+ ∑_{i+j+k=m} ∇^{i}ℓ ∗ ∇^{j}ℓ ∗ ∇^{k}ℓ ∗ ∇^{m}ℓ).

Estimating the terms on the RHS by their absolute value and using ab ≤ δa^{2} + (δ^{−1}/4)b^{2}, we then derive

|ℓ ∗ ^{H}∇_{H}∇^{m}ℓ ∗ ∇^{m}ℓ| ≤ |^{H}∇_{H}∇^{m}ℓ|^{2} + c(n, m)|ℓ|^{2}|∇^{m}ℓ|^{2},

such that

□_{ε}|∇^{m}ℓ|^{2} ≤ −|^{H}∇_{H}ℓ|^{2} + c(n, m)(|ℓ|^{2} + |∇ℓ|)|∇^{m}ℓ|^{2}

+c(n, m) ∑_{{i+j=m \atop i≤m−1,j≤m−2}} |∇^{i}ℓ|^{2}|∇^{j+1}ℓ|^{2}

+c(n, m) ∑_{{i+j+k=m \atop i,j,k≤m−1}} |∇^{i}ℓ|^{2}|∇^{j}ℓ|^{2}|∇^{k}ℓ|^{2}.

QED

Corollary 3.8(i) Let D = max_{N_{0}} |∇ℓ|^{2} and 0 ≤ ε ≤ 1. There are constants c_{3} > 0,

c_{4} > 0 depending only on c_{0}(n), c_{1}(n), c_{2}(n) defined previously such that max_{N_{t}} |∇ℓ|^{2} ≤

2max{D, B^{2}} for all 0 ≤ t ≤ t_{1} := min{c_{3}B^{−1}, c_{4}D^{−1/2}}. (ii) For each m ≥ 2 there is a constant c_{m} depending on n, m and max_{N_{0}} |∇^{i}ℓ|^{2}, 1 ≤ i ≤ m,

such that max_{N_{t}} |∇^{m}ℓ|^{2} ≤ c_{m} for all 0 ≤ t ≤ t_{1}.

Proof. (i) By Proposition 3.7 we have

□_{ε}|∇ℓ|^{2} ≤ c_{1}|∇ℓ|^{3} + c_{2}|ℓ|^{2}|∇ℓ|^{2}
≤ (c_{1} + c_{2}))|∇ℓ|^{3} + \frac{c_{2}B^{3}}{(1 − c_{0}Bt)^{3}}

by Young’s inequality and Corollary 3.6. Then |∇ℓ|^{2} is dominated by the comparison function g(t) = k_{1}^{2}/(1 − k_{2}t)^{2} if k_{1}^{2} = max\{D, B^{2}\}, k_{2} = max\{(2c_{0} + 2\sqrt{c_{2}})B, \sqrt{c_{1} + c_{2}}k_{1}\} and t_{1} = min\{c_{3}B^{−1}, c_{4}D^{− \frac{1}{2}} \} is chosen small enough. This gives the desired estimate. (ii) The higher derivative estimates follow from induction and Proposition 3.7(ii), since the interval is bounded. QED Lemma 3.9. Let 0 ≤ ε ≤ 1. Then for any solution of (d/dt) F = (L + εL^{N})X_{2n}, the induced metric ^{N}h on the hypersurfaces N_{t} is uniformly equivalent to the initial metric on N_{0} for 0 ≤ t ≤ 1/2 t_{0}. Proof: This follows from the boundedness of |ℓ|^{2} for 0 ≤ t ≤ 1/2 t_{0} as in [?] since |ℓ|^{2} controls the derivative of the induced metric in view of (2.1). QED Proof of Theorem 3.1.: Since the evolution equation (d/dt)F = (L + εL^{N})X_{2n} is strictly parabolic up to tangential diffeomorphisms for each 0 < ε ≤ 1, there is a smooth solution F^{ε} with initial data F_{0} for 0 < ε ≤ 1. In view of the Corollary and Lemma, each solution F^{ε} can be smoothly extended up to t_{1} = min{1/2 c_{0}^{−1}B^{−1}, c_{3}D^{−1/2}}. In view of these uniform estimates we may then pass ε → 0 to obtain the desired solution of (??). QED

4 Barrier principle

We demonstrate the parabolic nature of the flow (??) by showing that the distance between two disjoint surfaces is nondecreasing under the flow. In addition, embedded surfaces remain embedded.

Theorem 4.1. (i) If F_{j} : N^{j} × [0, t) → ℂ^{n}, j = 1, 2, are two smooth solutions of (??) with N_{0}^{1} ∩ N_{0}^{2} = ∅, then N_{t}^{1} ∩ N_{t}^{2} = ∅ for all t ∈ [0, T) and dist(N_{t}^{1}, N_{t}^{2}) does not decrease in time. (ii) If F_{0} : N → ℂ^{n} is embedded, then the solution F_{t} of (??) remains embedded for all t ∈ [0, T).

Corollary 4.2 If N_{0} ⊂ B(x_{0}, R) then N_{t} ⊂ B(x_{0}, R_{t}) for R_{t} = (R_{0}^{2} − 4(n − 1)t)^{1/2}. In particular, T < (4(n − 1))^{−1}R_{0}^{2}.

Proof of Theorem 4.1: For two hypersurfaces F_{j} : N × [0, T) → ℂ^{n}, j = 1, 2, evolving according to (??), we define the distance g : N × N × [0, T) → ℝ by g(p, q, t) = |F_{1}(p, t) − F_{2}(q, t)|^{2}. Then

\frac{∂}{∂t}g = 2(F_{1} − F_{2}, Ḟ_{1} − Ḟ_{2}) = 2(F_{1} − F_{2}, ^{H}ΔF_{1} − ^{H}ΔF_{2}).

Now let (p , q , t ) be an arbitrary point of N^{1} × N^{2} × [0, T), where g(p , q , t ) ≤ g(p, q, t) for all (p, q) ∈ N^{1} ×N^{2}. In view of [Ha], it is sufficient to show that (d/dt)g(p , q , t ) ≥ 0. In (p , q , t ) we have (F_{1} − F_{2})||X_{2n}. Choose a unitary frame {e_{α}^{1}}_{1}^{n−1} in H^{1,0}N_{t}^{1} at the point F_{1}(p , t ) and extend it parallel along the straight line from F_{1}(p , t ) to F_{2}(p , t ), where we get again a unitary frame {e_{α}^{2}}_{1}^{n−1} in H^{1,0}N_{t}^{2} . The first variation of g in direction of e_{ξ}^{1} ⊕ e_{ξ}^{2} is then given by

δg(e_{ξ}^{1} ⊕ e_{ξ}^{2}) = 2(F_{1} − F_{2}, ^{N}∇_{e_{ξ}^{1}}F_{1} − ^{N}∇_{e_{ξ}^{2}}F_{2}),

and the second variation is given by

δ^{2}g(e_{ξ}^{1}, e_{ξ}^{2}) ⊗ (e1 ; ξ̄, e2 ; ξ̄) = 2|e_{ξ}^{1} − e_{ξ}^{2}|^{2} + 2(F_{1} − F_{2}, ^{N}∇_{e1 ; ξ̄}e_{ξ}^{1} − ^{N}∇_{e2 ; ξ̄}e_{ξ}^{2}).

Since X_{2n} ∈ F_{j}^{∗}T ℂ^{n} is normal to T N,

(X_{2n}, ^{N}∇_{ej ; ξ̄}e_{ξ}^{j}) = (X_{2n}, ^{H}∇_{ej ; ξ̄}e_{ξ}^{j}) = (X_{2n} ,^{H} ΔF_{j}).

Therefore,

\frac{∂}{∂t}g(p, q, t) = δ^{2}g(e_{ξ}^{1} ⊗ e_{ξ}^{2}) ≥ 0.

This proves that the minimal distance between N_{t}^{1} and N_{t}^{2} does not decrease. To show the second part of the Theorem, note that from Theorem 3.1 we know that the solution remains smooth and regularly immersed as long as the curvature and its gradient remain bounded. By the argument above, g cannot attain a new interior minimum off the diagonal, which is the only place where it is zero initially by assumption. Let T > 0 be the maximal time of existence of F_{t} and δ > 0 be arbitrary. Then a uniform curvature bound |ℓ|^{2} ≤ B_{δ} exists on N × [0, T − δ) and g is uniformly bounded below on the boundary of a suitable small neighborhood of the diagonal in N × N for t ∈ [0, T − δ). Thus g remains nonzero off the diagonal on [0, T − δ). Since δ > 0 was arbitrary, the result follows. QED

5 Maximum principles

In order to establish qualitative properties of solution to (??) as well as their shape, we use the parabolic maximum principle for the operator \frac{d}{dt} −^{H}△ when applied to L and ℓ.

Theorem 5.1.

(i) −L ≥ 0 is preserved under the flow (??) (ii) If at time t = 0 we have −L ≥ c_{1} > 0 then −L has a lower bound given by

min_{N_{t}}(−L) ≥ \frac{c_{1}}{\sqrt{1 − tc_{1}^{2}/(n − 1)}}.

Proof: Since Proposition 3.4 implies

(\frac{d}{dt} −^{H}△) L = i\sqrt{2} (ℓ_{nγ̄}e_{γ} − ℓ_{nγ}eγ̄) · L + \frac{1}{2}L (ℓ_{γγ̄}^{2} + ℓ_{γ̄γ}^{2} + 2ℓ_{nn}^{2}) ,

the first result follows from the weak maximum principle for \frac{d}{dt} −^{H}Δ. Since ℓ_{γγ̄}^{2} ≥ \frac{1}{2n−2}L^{2}, we have (\frac{d}{dt} −^{H}Δ) (−L) ≥ i\sqrt{2} (ℓ_{nγ̄}e_{γ} − ℓ_{nγ}eγ̄) · (−L) − \frac{1}{2n−2}L^{3}. By comparison with the solutions of the ODE

\frac{d}{dt}f = \frac{1}{2n − 2}f^{3}, f(0) = c_{0},

the second part of the result follows. Note that equality holds on spheres. QED While we only use a weak version of the maximum principle in the last result, we now prove a strict Hopf-type parabolic maximum principle for pseudoconvex surfaces in ℂ^{2}. As a result, equation (1.1) provides a canonical deformation of smooth weakly pseudoconvex surfaces into strictly pseudoconvex surfaces in ℂ^{2}.

Theorem 5.2. Let F_{0} : N^{3} → ℂ^{2} be a smooth closed weakly pseudoconvex hypersurface. Then the solution F : N^{3} × [0, T) → ℂ^{2} of (1.1) on a maximal time interval has strictly pseudoconvex surfaces N_{t}^{3} for all 0 < t < T.

For the proof of Theorem 5.2 we need a Frobenius-type integrability theorem for certain distributions. Let U be a smooth real n-manifold and D^{m} ⊂ T U a smooth distribution in T U, or a subbundle of T U, with fibre dimension m < n, and fix p ∈ U. A piecewise regular real curve in U is called a D-curve if its tangent spaces lie in D. Now Γ(D) denotes the set of points in U which can be connected with p by a D-curve.

Theorem 5.3. Assume that D is involutive on Γ(D). Then in a neighborhood U_{p} of p there exist coordinates w : U_{p} ⟶ ℝ^{n}, w(p) = 0, with span{∂/∂w^{i}}_{1}^{m} = D on {w^{ℓ} = 0}_{m+1}^{n}.

Proof: The Theorem holds in the case that the fibre dimension m of D equals one. We will use induction to prove it for any m ≤ n − 1. Assume it holds for any distribution of fibre dimension m−1. Let D = span {X_{i}}_{1}^{m} and u : U_{p} ⟶ ℝ^{n}, u(p) = 0, be coordinates with ∂/∂u^{1} = X_{1}, and D_{1} = {X ∈ D : Xu^{1} = 0} ⊂ D. Now if X, X̃ ∈ D_{1}, then [X, X̃] ∈

D on Γ(D) and [X, X̃] u^{1} = 0. Therefore D_{1} ⊂ T U is involutive on Γ(D). Now let U_{1} := {u^{1} = 0} ⊂ U_{p}, then D_{1} | U_{1} ⊂ T U_{1} is involutive on Γ(D_{1}) ∩ U_{1}. By induction hypothesis, on some U_{2} ⊂ U_{1}, there exist coordinates (v^{2}, ⋯, v^{n}) : U_{2} → ℝ^{n−1}, v(p) = 0, with span {∂/∂v^{i}}_{2}^{m} = D_{1} | U_{2} on {v^{ℓ} = 0}_{m+1}^{n}. We can shrink U_{p} to define a map

π_{1} : U_{p} → U_{2}, π_{1}(q) = u^{−1} ∘ (0, u^{2}, ⋯, u^{n}) (q),

and new coordinates w : U_{p} ⟶ ℝ^{n}, w(p) = 0, by w^{1} = u^{1}, w^{j} = v^{j} ∘π, j = 2, ⋯, n. Since span{Y_{1} := X_{1}, Y_{i} := X_{i} − (X_{i} · u^{1}) X_{1}}_{2}^{m} = D, we have [Y_{i}, Y_{j}] = z_{ijk}Y_{k} on Γ(D) for functions z_{ijk} : Γ(D) → ℝ that are smooth on smooth segments of curves in Γ(D). Note that Y_{1}w^{ℓ} = 0 for ℓ = m + 1, ⋯, n, so for j = 2, ⋯, m, we have Y_{1} (Y_{j}w^{ℓ}) = [Y_{1}, Y_{j}] w^{ℓ} and by definition of z_{ijk}:

(ODE) Y_{1} (Y_{j}w^{ℓ}) = z_{ijk} (Y_{k} · w^{ℓ}) on Γ(D).

Since Y_{j} ∈ span {∂/∂v^{j}}_{2}^{m} and w^{ℓ} = v^{ℓ} on U_{2},

(IC) Y_{j}w^{ℓ} = 0 on U_{2}.

Denote by γ_{v^{2},...,v^{m}} the integral curve of Y_{1} passing through (v^{2}, ..., v^{m}, 0, ..., 0) ∈ U_{2} ∩ {v^{ℓ} = 0}_{m+1}^{n}. Then γ_{v^{2},...,v^{m}} is contained in Γ(D). The above system of ODEs along γ_{v^{2},...,v^{m}} for the functions (Y_{j} · w^{ℓ}) with initial condition (IC) on U_{2} ∩ {v^{ℓ} = 0}_{m+1}^{n} has the unique solution Y_{j} · w^{ℓ} = 0 on γ_{v^{2},...,v^{m}}. Therefore on {w^{ℓ} = 0}_{m+1}^{n} , span {∂/∂w^{i}}_{1}^{m} = span {Y_{i}}_{1}^{m}, and the latter equals D. QED

Proof of Theorem 5.2: Suppose there is p_{0} ∈ N, t ≥ 0 such that L(p_{0}, t_{0}) = 0. For coordinates z_{1} = x_{1} + ix_{2}, z_{3} = x_{3} + ix_{4} of ℂ^{2} let C_{4} be the largest x_{4}-value where L equals zero at t = t_{0}, i.e.

A := {q ∈ N|L(q, t_{0}) = 0} ⊂ {x_{4} ≤ C_{4}}, A ∩ {x_{4} = C_{4}} ≠ ∅.

Let p ∈ A ∩ {x_{4} = C_{4}}. Notice that the first order transport term in the evolution equation of L as given in the proof of Theorem 5.1 involves only horizontal derivatives of L and therefore has no component in the direction where ^{H}Δ is degenerate. The strict maximum principle for degenerate parabolic equations, see [?] and [?], then applies and shows that L has to be zero at each point (q, t_{0}) which can be reached from p by curves following a direction in HN_{t_{0}}, i.e. in Γ_{p}(HN_{t_{0}}) ∈ A. In particular, C ⊂ {x_{4} ≤ C_{4}} and p ∈ C ∩ {x_{4} ≤ C_{4}}. By the elliptic strong maximum principle for holomorphic curves it follows that C ⊂ {x_{4} = C_{4}} ∩ N_{t_{0}} and is a subset of an affine complex line in ℂ^{2}. Since A and N_{t_{0}} are closed, we could repeat the above argument at a possible boundary point of C, concluding that C can be extended in N_{t_{0}} to the entire affine complex line. This contradicts the compactness of N_{t_{0}}. QED

References

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[C] G. Citti, C^{∞} regularity of solutions of the Levi equation, Ann. Inst. H. Poincaré, Anal. non Linéaire 15,4 (1998) 517-534.

[Ha] R. Hamilton, Three-manifolds of positive Ricci curvature, J. Diff. Geom. 17 (1982) 255-306.

[H] G. Huisken, Flow by mean curvature of convex of convex surfaces into spheres, J. Diff. Geom. 20 (1984) 237 - 266.

[K] W. Klingenberg, Real hypersurfaces in Kähler manifolds, to appear in Asian Journal of Mathematics.

[PW] M. Protter, H. Weinberger, Maximum principles in differential equations, Prentice Hall, NJ, 1967.

[ST] Slodkowski, G. Tomassini, Evolution of subsets in C^{2} and a parabolic problem for the Levi equation, 1997, to appear in Annali della Scuola Normale di Pisa.

Mathematisches Institut Universität Tübingen Auf der Morgenstelle 10 72076 Tübingen Germany gerhard.huisken@uni-tuebingen.de

Institut für Reine Mathematik Humboldt Universität zu Berlin Ziegelstrasse 13A 10099 Berlin Germany klingenb@mathematik.hu-berlin.de