On the functional ∫_Ωf + ∫_Ω*g

2.1 The Minkowski problem in the sphere

This problem can be phrased as follows. Given a positive function ϕ on the unit sphere S n , one asks whether there exists a convex hypersurface M ⊂ S n such that the Gauss curvature of M at a point x ∈ M is equal to ϕ(ν(x)),

where ν(x) is the unit outer normal of M at x.

This Minkowski problem can be reformulated as a variational problem associated with the functional (1.3), subject to the constraint

Let Ω be a domain in S n . The polar dual of Ω is a convex set in S n given by

For simplicity we will consider domains with Lipschitz boundaries only, and we also assume that f, g are positive and bounded measurable functions on S n .

Given a domain Ω and a constant δ > 0, we denote by N δ ( Ω ) the union of subsets of S n whose Hausdorff distance from Ω is less then δ, namely N δ ( Ω ) = Ω ′ ⊂ S n : Ω ⊂ Ω δ ′ and Ω ′ ⊂ Ω δ , where Ω δ = { x ∈ S n : dist ( x , Ω ) < δ } .

When Ω and Ω* are dual to each other, we call them a dual pairing. The duality on the unit sphere has a special property, namely the unit outer normal ν(x) of x ∈ ∂Ω is a point on ∂Ω*, and for any point y ∈ ∂Ω*, there is a point x ∈ ∂Ω such that y = ν(x). If ∂Ω and ∂Ω* are C ¹ smooth and strictly convex, then ν: x ∈ ∂Ω → ν(x) ∈ ∂Ω* is a one-to-one mapping on ∂Ω. Observe that if ∂Ω is strictly convex and C ¹ smooth, so is ∂Ω*.

Let Ω , Ω * ⊂ S n be a dual pairing. Denote M = ∂Ω and M* = ∂Ω*. Let κ _i be the principal curvatures of M at x ∈ M, where i = 1, …, n − 1. Then κ i − 1 are the principal curvatures of M* at y = ν(x). Hence if M is a solution to the Minkowski problem (2.1), then M* is a solution to the prescribed Gauss curvature problem

Therefore, problem (2.4) is equivalent to its dual problem (2.1).

Next we derive the Euler equation for the functional J .

When f ≡ 1 and g = 1/ϕ, equation (2.5) is the Minkowski problem in the sphere, namely equation (2.1). The Minkowski problem in the sphere was studied by Gerhardt [12]–[14]. He proved the existence of one solution if certain barrier conditions are satisfied or ϕ is invariant under a group action [12]. In the paper [2], by exploiting the functional (1.3), we proved the existence of solutions to the Minkowski problem (2.1) in the sphere for any positive and bounded function ϕ.

2.2 Extensions

We can extend the functional (1.3) from the sphere S n to more general convex hypersurfaces or even manifolds. Here are two examples.

The above are two examples of the functional (1.3). We point out that there are more functionals like the functional J , which also leads to interesting Monge-Ampère type equations.

We now consider an extension of (1.3)

where M is a closed convex hypersurface, M* is the polar dual of M, κ * = κ 1 * , … , κ n * are the principal curvatures of M* and ν* is the unit outer normal of M*. Let K, K* be the Gauss curvature of M, M*, respectively. Then d s = d ν K , d s * = d ν * K * . Hence

where λ = (λ ₁, …, λ _n ) and λ * = λ 1 * , … , λ n * are the principal radii of M and M*. Let u, u* be the support functions of M, M*. Then

where F(ν) = Δf + nf and G(ν*) = Δg + ng. Let M _t be a variation of M and M t * be its dual. Denote u(⋅, t) and u*(⋅, t) the support functions. Recall that (see e.g. [16])

where r* is the radial function of M*, and ν is the unit outer normal of M at the point p such that p/|p| = ν*. Therefore

where we use d ν * = u r n + 1 K d ν and u* = 1/r in the last equality (see e.g. [16]). Here r is the radial function of M. As the variation ∂ _t u can be arbitrary, we see that the Euler equation of the functional I is

where ( u 2 + | ∇ u | 2 ) 1 2 ( ν ) = r ( ν * ) is used. Here K(ν) means the Gauss curvature K at the point p ∈ M such that the unit outer normal of p is ν. Equation (2.15) is a L _p dual Minkowski problem [17].

3.1 Existence of one solution

In [2], we introduced a piece-wise smooth gradient flow for the functional J , for which we can establish the a priori estimates and prove the convergence. Together with the Mountain Pass Lemma, we proved the existence of one solution to the Minkowski problem in the sphere.

Let us recall the Mountain Pass Lemma, as a comparison to the argument below for the second solution. Let I be a functional defined on a metric space H. Let L be the set of all paths connecting two fixed points p ₀, p ₁ ∈ H, namely

Assume that there exists a constant δ ₀ > 0 such that ∀ ℓ ∈ L,

Let

Then under appropriate conditions, c* is a critical value of I .

To find a critical point, one employs an ascent (or descent) gradient flow G : p ∈ H → p ; t ∈ H , such that for any ℓ = {p _s : s ∈ [0, 1])} ∈ L and t > 0, the flow G sends ℓ to ℓ _;t  = {p _s;t : s ∈ [0, 1]} ∈ L. Under appropriate conditions, one can prove that there exists s* ∈ (0, 1), such that p s * ; t converges to a critical point p* of I and I ( p * ) ≥ c * .

3.2 Existence of two solutions

A new topological method was introduced in [2] to prove that for any bounded, positive function ϕ on S n + 1 , there exist at least two solutions to the Minkowski problem in the sphere.

In the case f = g ≡ 1, the dual pairing

is a critical point of J , and it is the Mountain Pass solution of the functional J . Any other solution must be a translation of Ω, Ω* given above [12]. Hence in our Theorem 3.2 below, when ϕ is a constant, two geodesic spheres S n with different centres are regarded as different solutions. Hence when f = g ≡ 1, there are infinitely many solutions, and these solutions are congruent with each other by translations.

Our argument is also a min-max principle like the Mountain Pass Lemma. In Theorem 3.1, we obtain one solution M* to the Minkowski problem in the sphere by the min-max principle (3.4). To obtain the second solution, we use the gradient flow (3.2) and replace the path ℓ in (3.4) by cylinder C , and obtain a different critical point for the functional J in (1.3).

More precisely, let Θ denote the set of all closed convex hypersurfaces in S n (a single point p ∈ S n will also be regarded as an element in Θ). A cylinder C is a mapping from S n × [ 0,1 ] to Θ such that

1. C ( p , 0 ) = { p } for any point p ∈ S n and C ( p , s ) shrinks to the point {p} as s → 0.

2. C ( p , s ) is continuous in p ∈ S n and s ∈ [0, 1].

3. C ( p , 1 ) = ∂ B π / 2 ( p ) , where B r ( p ) ⊂ S n is the geodesic ball with centre p and radius r.

By the above properties, one sees that C ( S n × [ 0,1 ] ) is a topological cylinder.

Let Φ be the set of all cylinders. We denote

where J is the functional (1.3) with f ≡ 1, g = 1/ϕ. For simplicity we abbreviate J ( Ω , Ω * ) to J ( Ω ) when Ω* is the polar dual of Ω, and for convenience we also write J ( Ω ) as J ( ∂ Ω ) .

By definition, we have c* ≥ c _*, where c* is the critical value given in (3.4) for the functional J in (1.3). Corresponding to (3.3), it is easy to verify that

To obtain the second critical point, we use the flow (3.2) to deform an initial cylinder C 0 to C t , such that for any ( p , s ) ∈ S n × [ 0,1 ] , C t ( p , s ) ∈ Θ is the solution to the flow with initial condition C 0 ( p , s ) . For any t > 0, by the definition of c*, we have

Naturally we choose the initial cylinder C 0 by C 0 ( p , s ) = ∂ B s π / 2 ( p ) , the geodesic sphere centred at p with radius sπ/2. But one can also choose any cylinder C ∈ Φ as the initial condition.

Note that C t ( p , s ) , as a convex hypersurface in S n , has a unique geometric centre. It is a point on S n depending continuously on p, s and t. Therefore for any s ∈ (0, 1) and t > 0, the set of geometric centres of C t ( p , s ) is a cover of the unit sphere S n , which cannot continuously deform to a point in S n . This is a key topological property in this argument.

By this property and the definition of c _*, there exists ( p * , s * ) ∈ S n × ( 0,1 ) such that J ( C t ( p * , s * ) ) ≤ c * for all t > 0. By the a priori estimates, C t ( p * , s * ) converges to a critical point M _* of J with J ( M * ) ≤ c * .

If J ( M * ) < c * , we can choose another cylinder C ′ with inf ( p , s ) ∈ S n × [ 0,1 ] J ( C ′ ( p , s ) ) > J ( M * ) , and repeat the above procedure to obtain another solution M * ′ with J M * ′ > J ( M * ) . By the a priori estimates for equation (3.1) and the gradient flow, one sees that c _* is a critical value of J .

If c* ≠ c _*, apparently the two solutions corresponding to the critical values c* and c _* are different. If c* = c _*, we can show that there are two different solutions, using the nontrivial topology of S n , i.e. it is not contractible [2].

We also show that there exists positive functions ϕ ∈ C 2 ( S n ) such that the Minkowski problem (3.1) admits exactly two solutions. Such a function ϕ we gave in [2] is rotationally symmetric with respect to the x _n+1 axis and is strictly monotone in x _n+1. By the rotating plane method, we show that a solution must be a geodesic sphere centred at the north pole or the south pole. The rotating plane method is similar to the moving plane method. On the unit sphere, we can rotate a plane passing through the x _n+1-axis and use it to prove the symmetry.

4.1 The centro-affine Minkowski problem

Denote by K o the set of convex bodies in R n + 1 enclosing the origin. In this section we deal with the existence of critical points Ω , Ω * ∈ K o of the functional

subject to the constraint x ⋅ y ≤ 1 ∀ x ∈ Ω, y ∈ Ω*. By the proof of Lemma 2.1, if the pairing Ω, Ω* is a critical point of J , then necessarily Ω is convex and Ω* is the polar dual of Ω, namely

Hence as before, we may abbreviate J ( Ω , Ω * ) to J ( Ω ) for simplicity.

Assume that Ω ∈ K o is a critical point of J . Denote by u the support function of M = ∂Ω. Then u satisfies the Euler equation of J ,

where I is the identity matrix, and ∇ denotes the covariant derivative with respect to an orthonormal frame on S n .

When f ≡ 1, equation (4.2) is precisely the prescribed centroaffine curvature problem studied in [18], which is a variant of the centroaffine Minkowski problem in [19]. Recall that the centroaffine curvature (introduced by Tzitzéica [20] in 1908) of a convex hypersurface M ⊂ R n + 1 at a point p ∈ M is equal to the quantity

where K(p) is the Gauss curvature of M at p, d(p) is the distance from the origin to the tangent plane of M at p, and x is the unit outer normal of M at p.

The centroaffine Minkowski problem concerns the existence of closed convex hypersurfaces M such that the centroaffine curvature is equal to a given function g. If g is a function of the centro-affine normal of M, it is the centroaffine Minkowski problem introduced in [19]. If g is a function of the position of M, it is the version studied in [18]. We remark that for the prescribed Gauss curvature equation K(p) = g(p) ∀ p ∈ M, where g is a function of the position of M, the problem has also been studied in a number of papers [21]–[23].

Due to the affine invariance, there is no uniform estimate for equation (4.2) for general functions f and g. For example, when f = g = 1, (4.2) is the equation for elliptic affine spheres. A classical result in affine geometry states that a closed convex hypersurface is an elliptic affine sphere if and only if it is an ellipsoid. Hence ellipsoids with the volume |B ₁(0)| are the only critical points of the functional J .

In the paper [1], we proved the following existence of solutions to (4.2).

There is no loss of generality in assuming g(z) → ∞ as |z| → ∞. By our proof of Theorem 4.1, one sees that this condition can be relaxed to g ≥ C when |z| is sufficiently large, for a suitably large constant C > 0. To prove Theorem 4.1, we will use the following Gauss curvature flow

together with a topological method, where X(⋅, t) is a parametrisation of the evolving convex hypersurfaces M _t , ν is the unit outward normal, K is the Gauss curvature of M _t , and ⟨X, ν⟩ denotes the inner product of X, ν in R n + 1 .

The main difficulty is the uniform estimate. As shown by the example f ≡ 1 and g ≡ 1, there is no uniform estimate in general. We will use a topological method to find a special initial hypersurface such that the Gauss curvature flow (4.3) is uniformly bounded. This topological method was first introduced in our previous work [5], to prove the existence of a solution to the L _p -Minkowski problem in the super-critical exponent case.

Let M _t be a solution to the flow (4.3) and let u(⋅, t) be the support function of M _t . Then the flow (4.3) can be expressed as

One can verify that d d t J ( Ω t ) ≥ 0 , and the equality holds if and only if M _t satisfies (4.2), where Ω _t  = Cl(M _t ) is the convex body enclosed by M _t .

Moreover, if the solution satisfies the uniform estimate

for all time t ≥ 0, then we have the estimate for the second derivatives,

where the constant C depends only on n, C ₀, f, g and the initial condition u(⋅, 0).

By the second derivative estimates, equation (4.4) becomes uniform parabolic. Hence by Krylov’s regularity theory, higher regularity also follows:

∀ α ∈ (0, 1), where C depends only on n, α, C ₀, f, g and the initial condition u(⋅, 0).

4.2 The L _p -Minkowski problem

This problem can be formulated as solving the equation

where f is a positive function on the sphere S n , u is the support function of a closed convex hypersurface M ⊂ R n + 1 . Equation (4.8) includes the classical Minkowski problem (p = 1), the logarithmic Minkowski problem (p = 0), and the centro-affine Minkowski problem (p = −n − 1) as special cases [8], [19].

The L _p -Minkowski problem has been extensively studied in the last three decades, after it was introduced in [24]. According to the Blaschke-Santaló inequality, the problem is divided into three cases, namely the subcritical growth case p > − n − 1, the critical case p = −n − 1 and the super-critical case p < − n − 1. In the subcritical case, the existence of solutions can be obtained by using the variational method and the Blaschke-Santaló inequality [19]. In the critical case, a Kazdan-Warner type obstruction was found in [19]. It implies that (4.8) admits no solution for a general positive function f. But in this case, the problem may also have infinitely many solutions for some f [25]. In the supercritical case, there is no uniform estimate, and one might believe that the problem has no solution. Surprisingly, we prove that for any positive f, equation (4.8) has a solution. The key to proving this existence result is the topological method to be introduced in the next section. We will discuss the method for equation (4.2). For equation (4.8), interested readers are referred to [5] for details. We have also applied this method to the L _p dual Minkowski problem [6].

5.1 A property of the functional

Given a convex body Ω ∈ K o , let r ₁(Ω) ≤ r ₂(Ω) ≤ … ≤ r _n+1(Ω) be the lengths of semi-axes of the minimum ellipsoid E(Ω). Let us define the eccentricity of Ω by

Then we have the following property [1]:

5.2 A modified flow of (4.3)

Fix the constant

If the minimum ellipsoid of Ω ∈ K o is the unit ball B ₁, then we have J ( Ω ) ≤ 1 2 A 0 .

For a smooth and uniformly convex hypersurface N with Ω 0 = Cl ( N ) ∈ K o , we define a modified flow M ̄ N ( t ) with initial condition N as follows:

1. If J ( N ) < A 0 , we deform N by the flow (4.3) and let M _N (t) be the solution. As the functional J ( M N ( t ) ) is non-decreasing, if J ( M N ( t ) ) reaches A ₀ at the first time t′, we stop the flow at time t = t′ and freeze the solution thereafter. That is, M ̄ N ( t ) = M N ( t ) for 0 ≤ t < t′ and M ̄ N ( t ) = M N ( t ′ ) for all t ≥ t′.

2. If J ( N ) ≥ A 0 , we set M ̄ N ( t ) ≡ N for all t ≥ 0, i.e., the solution is stationary.

Therefore M ̄ N ( t ) is defined for all time t ≥ 0, and J ( M ̄ N ( t ) ) is non-decreasing. By Lemma 5.1, if one of Vol(Ω₀), [ Vol ( Ω 0 ) ] − 1 , [ dist ( O , ∂ Ω 0 ) ] − 1 , and e Ω 0 is sufficiently large, then we have M ̄ N ( t ) ≡ N for all t.

We want to find an initial condition N such that J ( M N ( t ) ) < A 0 for all time t > 0 and its support function u satisfies the uniform estimate (4.5). If this is done, then by the a priori estimates (4.6) and (4.7), M _N (t) converges to a solution of (4.2). We prove the existence of such an initial condition N by the following topological method.

5.3 Homology for a class of ellipsoids

Fix the constant A ₀ as above. By Lemma 5.1, there exist small constants d ̄ and v ̄ , and large constant e ̄ , such that J ( Ω ) ≥ A 0 provided Ω ∈ K o satisfies

Let

where v 0 = v ̄ , v 1 = ( n + 1 ) n + 1 v ̄ − 1 . Equipped with the Hausdorff distance, A I is a subspace of the metric space K of all convex bodies in R n + 1 . One can verify that

1. A I is contractible and so homology group H k ( A I ) = 0 for all k ≥ 1. Denote

   P = { E ∈ A I : either  Vol ( E ) = v 0 ,  or  Vol ( E ) = v 1 ,  or  e ( E ) = e ̄ ,  or  dist ( O , ∂ E ) = 0 } .

   One can verify

2. There exists a retraction Ψ: A I \ { B 1 } → P , i.e., Ψ : A I \ { B 1 } → P is continuous and Ψ | P = i d , where B 1 = B 1 ( 0 ) ∈ K o is the unit ball. Denote

   E = { E ∈ A I ∩ K e : Vol ( E ) = Vol ( B 1 ) , e ( E ) = e ̄ } , A = { E ∈ A I : Vol ( E ) = Vol ( B 1 )  and either  e ( E ) = e ̄  or  dist ( O , ∂ E ) = 0 } ,

   where K e denotes the set of the origin symmetric convex bodies. Then we also have

3. H k + 1 ( P ) = H k ( A ) for all k ≥ 1,

4. There is a long exact sequence

   … → H k + 1 ( A ) → H k ( E × S n ) → H k ( E ) ⊕ H k ( S n ) → H k ( A ) → … ,

5. The (n* + n − 1)-th homology group of E satisfies

   H n * + n − 1 ( E ) = Z , where  n * = n ( n + 1 ) 2 .

   For the proof of (a ₁)–(a ₅), we refer the reader to [5].