A curious identity on the median triangle

I just came across a curious identity about the angles of the “median triangle” of a given triangle, while I was reviewing a paper from a team participating in the Hang Lung Mathematics Award. Of course, I am not going to reveal the identity for the obvious reason.

Let me first describe the setting. Let {\Delta ABC} be a triangle. (By abuse of notations, we regard (for example) {A} both as a vertex, a vector (in {\mathbb R^2} or {\mathbb R^3}), and the angle of the triangle {\Delta ABC} at the vertex {A}.) We can then draw the three medians on the triangle which, as is well-known, intersect at the so called centroid of the triangle. Let {D, E}, and {F} be the angles at the centroid as shown:triangle1.png

Theorem 1 We have the identity

\displaystyle \frac{1}{\sin^2 D}+ \frac{1}{\sin^2 E}+ \frac{1}{\sin^2 F} =\frac{1}{\sin^2 A}+ \frac{1}{\sin^2 B}+ \frac{1}{\sin^2 C}.

This result certainly looks very elegant (and is new to me). However, the proof in that paper consists of several pages of computations which to me is not very enlightening. So I set out to write a proof myself, which will be described below. Nevertheless, I have to resort to coordinates to prove the result. It would be desirable to know if there is a more classical proof without using coordinates. (Of course, all the computations using coordinates can theoretically be translated to classical statements, e.g. the cosine law is just the expansion of the inner product { |A-B|^2}. However, I am not quite willing to do such kind of line-by-line translation. ) Continue reading

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Posted in Geometry | 4 Comments

27 lines on a smooth cubic surface

Here describes two different proofs of a general smooth cubic surface containing exactly 27 lines. One approach uses blow-ups and the other one uses the Grassmannian.

If I have time I will elaborate on the discussion.

 

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Weighted Hsiung-Minkowski formulas and rigidity of umbilic hypersurfaces

1. Motivation and Main Results

A. D. Alexandrov [Ale1956], [Ale1962] proved that the only closed hypersurfaces of constant (higher order) mean curvature embedded in {{\mathbb{R}}^{n \geq 3}} are round hyperspheres. The embeddedness assumption is essential. For instance, {{\mathbb{R}}^{3}} admits immersed tori with constant mean curvature, constructed by U. Abresch [Abr1987] and H. Wente [Wen1986] . R. C. Reilly [Rei1977] and A. Ros [Ros1987], [Ros1988] presented alternative proofs, employing the Hsiung-Minkowski formula. See also Osserman’s wonderful survey [Oss1990] .

In 1999, S. Montiel [Mon1999] established various general rigidity results in a class of warped product manifolds, including the Schwarzschild manifolds and Gaussian spaces. Some of his results require the additional assumption that the closed hypersurfaces are star-shaped with respect to the conformal vector field induced from the ambient warped product structure. As a corollary [Mon1999][Example 5] , he also recovers Huisken’s theorem [Hui1990] that the closed, star-shaped, self-shrinking hypersurfaces to the mean curvature flow in {{\mathbb{R}}^{n \geq 3}} are round hyperspheres. In 2016, S. Brendle [B2016] solved the open problem that, in {{\mathbb{R}}^{3}}, closed, embedded, self-shrinking topological spheres to the mean curvature flow should be round. The embeddedness assumption is essential. Indeed, in 2015, G. Drugan [Dru2015] employed the shooting method to prove the existence of a self-shrinking sphere with self-intersections in {{\mathbb{R}}^{3}}.

In 2001, H. Bray and F. Morgan [BM2002] proved a general isoperimetric comparison theorem in a class of warped product spaces, including Schwarzschild manifolds. In 2013, S. Brendle [B2013] showed that Alexandrov Theorem holds in a class of sub-static warped product spaces, including Schwarzschild and Reissner-Nordstrom manifolds. S. Brendle and M. Eichmair [BE2013] extended Brendle’s result to the closed, convex, star-shaped hypersurfaces with constant higher order mean curvature. See also [Gim2015] by V. Gimeno, [LX2016] by J. Li and C. Xia, and [WW2016] by X. Wang and Y.-K. Wang.

In this post, we provide new rigidity results (Theorem 1, 2 and 3). First, we associate the manifold {M^{n \geq 3} = \left( {N}^{n-1} \times [0,\bar{r}), \bar{g} = dr^2 + h(r)^2 \, g_ {N} \right)}, where {(N^{n-1}, g_N)} is a compact manifold with constant curvature {K}. As in [B2013], [BE2013] , we consider four conditions on the warping function {h: [0,\bar{r}) \rightarrow [0, \infty)}:

  • (H1) {h'(0) = 0} and {h''(0) > 0}.
  • (H2) {h'(r) > 0} for all {r \in (0,\bar{r})}.
  • (H3) {2 \, \frac{h''(r)}{h(r)} - (n-2) \, \frac{K - h'(r)^2}{h(r)^2}} is monotone increasing for {r \in (0,\bar{r})}.
  • (H4) For all {r \in (0,\bar{r})}, we have {\frac{h''(r)}{h(r)} + \frac{K-h'(r)^2}{h(r)^2} > 0}.

Examples of ambient spaces satisfying all the conditions include the classical Schwarzschild and Reissner-Nordstrom manifolds [B2013] [Section 5].

Theorem 1 Let {\Sigma} be a closed hypersurface embedded in {{ M }^{n \geq 3}} with the {k}-th normalized mean curvature function {H_{k}=\eta(r)>0} on {\Sigma} for some smooth radially symmetric function {\eta(r)}. Assume that {\eta(r)} is monotone decreasing in {r}.

  1. {k=1:} Assume (H1), (H2), (H3). Then {\Sigma} is umbilic.
  2. {k \in \{2, \cdots, n-1\}:} Assume (H1), (H2), (H3), (H4). If {\Sigma} is star-shaped (Section 2), then it is a slice {{N}^{n-1} \times \left\{ r_{0} \right\}} for some constant {r_{0}}.

 

We also prove the following rather general rigidity result for linear combinations of higher order mean cuvatures, with less stringent assumptions on the ambient space.

Theorem 2 Suppose {(M^{n \geq 3}, \bar g)} satisfies (H2) and (H4). Let {\Sigma} be a closed star-shaped {k}-convex ({H_k>0}) hypersurface immersed in {M^n}, {\{a_i(r)\}_{i=1}^{l-1}} and {\{b_j(r)\}_{j=l}^k} ({2\le l<k\le n-1}) be a family of monotone decreasing, smooth, non-negative functions and a family of monotone increasing, smooth, non-negative functions respectively (where at least one { a_i(r) } and one { b_j(r) } are positive). Suppose

\displaystyle \sum_{i=1}^{l-1}a_i(r)H_i= \sum_{j=l}^{k} b_j(r) H_j.

Then {\Sigma} is totally umbilic.

Theorem 2 contains the case where {\frac{H_k}{H_l}=\eta(r)} for some monotone decreasing function {\eta} and {k>l}. We notice that the same result also applies to the space forms {\mathbb R^n}, {\mathbb H^n} and {\mathbb S^n_+} (open hemisphere) without the star-shapedness assumption (Theorem 11). Our result extends [Koh2000][Theorem B] by S.-E. Koh, [Kwo2016][Corollary 3.11] by Kwong and [WX2014] [Theorem 11] by J. Wu and C. Xia.

We next prove, in Section 4, a rigidity theorem for self-expanding soliton to the inverse curvature flow. Let us first recall the well-known inverse curvature flow of hypersurfaces

\displaystyle \frac{d}{dt} \mathcal{F} =\frac{ {\sigma}_{k-1} }{ {\sigma}_{k} } \nu, \ \ \ \ \ (1)

 

where {\nu} denotes the outward pointing unit normal vector field and {{\sigma}_{k}} the {k}-th symmetric function of the principal curvature functions. We point out that the inverse curvature flow has been used to prove various geometric inequalities and rigidities: Huisken-Ilmanen [HI2001] , Ge-Wang-Wu [GWW2014] , Li-Wei-Xiong [LWX2014], Kwong-Miao [KM2014] , Brendle-Hung-Wang [BHW2016] , Guo-Li-Wu GLW2016 , and Lambert-Scheuer [LS2016] .

In the Euclidean space, the long time existence of smooth solutions to (1) was proved by Gerhardt in [G1990] and by Urbas in [U1990] , when the initial closed hypersurface is star-shaped and {k}-convex {({\sigma}_{k}>0)}. Furthermore, they showed that the rescaled hypersurfaces converge to a round hypersphere as { t \rightarrow \infty}.

Theorem 3 Let {\Sigma} be a closed hypersurface immersed in {{\mathbb{R}}^{n \geq 3}}. If it becomes a self-expanding soliton to the inverse curvature flow, it must be round.

In the proof of our main results, we shall use several integral formulas and inequalities. Theorem 1 requires the embeddedness assumption as in the classical Alexandrov Theorem and is proved for the space forms in [Kwo2016] . Theorem 2 and 3 require no embeddedness assumption and Theorem 3 is proved in [DLW2015] for the inverse mean curvature flow. Continue reading

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The discrete Gauss-Bonnet theorem

This is a slight extension of my previous note on discrete Gauss-Bonnet theorem. As mentioned in that note, this is a generalization of the well-known fact that the sum of the exterior angles of a polygon is always {{2\pi}}, which can also be regarded as a very special case of the Gauss-Bonnet theorem.

First of all we introduce a discrete version of surfaces. Let { M} be a ({2}-dimensional) “discrete surface” (here for examples) or a simplicial surface (with or without boundary), which for simplicity is assumed to be contained in {\mathbb R^n}. A discrete surface consists of finitely many “triangles” gluing along their edges. Each triangle is isometric to an ordinary planar triangle on {\mathbb R^2}. We call each triangle a “face”. For each face, there are exactly three edges. And for each edge, there are exactly two vertices. If two different faces intersect, we assume they either intersect at their common edge or their common vertex. Clearly the discrete notion of surfaces is very useful in computer graphics.

mesh surface

A discrete surface

discrete_surface

A more complicated discrete surface

The main result of this post is

Theorem 1 (Discrete Gauss-Bonnet theorem) Let {K} be the Gaussian curvature and {k_g} be the geodesic curvature on a discrete surface {M}, then

\displaystyle \displaystyle \sum_{v\in \boldsymbol{V_I}} K(v)+\sum_{v\in \boldsymbol{V_B}}k_g(v)=2\pi\chi(M).

Here {\boldsymbol{V_I}} is the set of interior vertices and {\boldsymbol{V_B}} is the set of boundary vertices and {\chi(M)} is the Euler characteristic of {M}.

We will explain the notation below the fold.

Our first goal is to generalize all the notions such as boundary, geodesic curvature (or exterior angle at an vertex of a triangle), and Gaussian curvature on a smooth surface to a discrete surface. Continue reading

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Why a vector field rotates about its curl?

Let {F= F(x_1, x_2, x_3): \mathbb R^3\rightarrow \mathbb R^3} be a smooth vector field on {\mathbb R^3}. We define its curl to be the vector field

\displaystyle \textrm{curl } F:= \left(\frac{\partial F_3}{\partial x_2}- \frac{\partial F_2}{\partial x_3}, \frac{\partial F_1}{\partial x_3}- \frac{\partial F_3}{\partial x_1}, \frac{\partial F_2}{\partial x_1}- \frac{\partial F_1}{\partial x_2} \right) \ \ \ \ \ (1)

curl

It is often claimed in advanced calculus that {\textrm{curl }F(p)} measures how fast {F} “rotates”. More specifically, suppose {\textrm{curl }F(p)\ne 0}, then locally around {p}, the vector field {F} behaves like a “swirl”, rotating (in the anti-clockwise sense) around the axis pointing in the same direction as {\textrm{curl }F(p)} and with angular speed {|\textrm{curl }F(p)|}. (Strictly speaking this picture is not quite true even locally: {\textrm{curl } F(p)} only captures the “rotational” part of {F(x)} around {p}. Besides the constant part {F(p)} and the “rotational” part {\textrm{curl }F(p)}, its first order part also contain the “flux” part which is given by {\textrm{div } F(p)}, but let’s ignore this for the discussion here.)

To my dismay, while this gives a good geometric description of {\textrm{curl }F}, I find that in many textbooks the demonstration of this “fact” is either by looking at the curl of some very special vector field (like {(-y, x, 0)}), or using the Stokes’ Theorem. The first approach is really too simplistic although it gives a good model for a “rotational” vector field. The second approach, while rigorous, is “a posteriori”, as it seems that the rather obscure definition of curl given in (1) just comes from nowhere. How on earth would one come up with such a definition at the very beginning? Continue reading

Posted in Calculus, Linear Algebra | 1 Comment

A functional inequality on the boundary of static manifolds

1. introduction and statement of results

The research in this article is largely motivated by the following result concerning a functional inequality on the boundary of bounded domains in the Euclidean space {\mathbb R^n}, proved in [MTX] Corollary 3.1.

Theorem 1 ([MTX]) Let { \Omega \subset \mathbb R^n } be a bounded domain with smooth boundary {\Sigma}. Let {H} and {\mathbb{II}} be the mean curvature and the second fundamental form of {\Sigma } with respect to the outward normal respectively. If {H>0}, then

\displaystyle \int_\Sigma \left[ \frac{ ( \Delta_{_\Sigma} \eta )^2 }{ H } - \mathbb{II} ( \nabla_{_\Sigma} \eta ,\nabla_{_\Sigma} \eta )\right] d \sigma \ge 0 \ \ \ \ \ (1)

 

for any smooth function { \eta } on {\Sigma}. Here { \nabla_{_\Sigma} }, { \Delta_{_\Sigma} } denote the gradient, the Laplacian on { \Sigma} respectively, and { d \sigma } is the volume form on { \Sigma }. Moreover, equality in (1) holds for some { \eta } if and only if { \eta = a_0 + \sum_{i=1}^n a_i x_i } for some constants {a_0, a_1, \ldots, a_n }. Here {\{ x_1, \ldots, x_n\}} are the standard coordinate functions on {\mathbb R^n}.

When {n=3} and {\Sigma} is convex, it is known ([MTX]) that the functional on the left side of (1) represents the second variation along {\eta} of the Wang-Yau quasi-local energy ( [WY1], [WY2] ) at the {2}-surface { \Sigma }, lying in the time-symmetric slice {\mathbb R^3 = \{ t = 0 \}}, in the Minkowski spacetime { \mathbb R^{3,1}}. Thus, (1) can be relativistically interpreted as the stability inequality of the Wang-Yau energy at { \Sigma}. The general case of such a stability inequality is implied by results in [CWY], [WY2] for a closed, embedded, spacelike {2}-surface in { \mathbb R^{3,1}} that projects to a convex {2}-surface along some timelike direction.

In this article, adopting a Riemannian geometry point of view, we generalize Theorem 1 to hypersurfaces that are boundaries of bounded domains in a simply connected space form. More generally, we give an analogue of (1) on the boundary of compact Riemannian manifolds whose metrics are static (see Definition 3).

First, we fix some notations. Given a constant {\kappa > 0}, let {\mathbb H^n (\kappa)} and {\mathbb{S}^n_+ ( \kappa )} denote an {n}-dimensional hyperbolic space of constant sectional curvature {-\kappa} and an {n}-dimensional open hemisphere of constant sectional curvature {\kappa } respectively.

Theorem 2 Suppose {(M , g)} is one of {\mathbb{R}^n}, {\mathbb H^n (\kappa)} and {\mathbb{S}^n_+(\kappa)}. Let {V } be the positive function on {M} given by

\displaystyle V= \begin{cases} 1, \quad & \displaystyle \mathrm{if} \ (M, g) =\mathbb{R}^n,\\ \cosh \sqrt\kappa r, \quad & \displaystyle \mathrm{if} \ (M, g) =\mathbb{H}^n(\kappa), \\ \cos \sqrt\kappa r, \quad & \displaystyle \mathrm{if} \ (M, g) =\mathbb{S}^n_+(\kappa), \end{cases} \ \ \ \ \ (2)

 

where {r} is the distance function from a fixed point {p} on {(M, g)}. When { (M, g) = \mathbb{S}^n_+ ( \kappa )}, {p} is chosen to be the center of {\mathbb{S}^n_+ ( \kappa ) } so that {V>0} on {M}. Given a bounded domain {\Omega \subset M } with smooth boundary {\Sigma}, let {H} and {\mathbb{II}} be the mean curvature and the second fundamental form of {\Sigma} respectively. If {H > 0 }, then for any smooth function {\eta} on {\Sigma},

\displaystyle \begin{array}{rl} & \displaystyle \int_\Sigma V \left[ \frac{ \left[ \Delta_{\Sigma} \eta + (n-1) k \eta \right]^2 }{H} - \mathbb{II} (\nabla_{\Sigma} \eta, \nabla_{\Sigma} \eta) \right] d \sigma \\ \ge & \displaystyle \int_\Sigma \frac{\partial V}{\partial \nu} \left[ | \nabla_{\Sigma} \eta |^2 - (n-1) k \eta^2 \right] d \sigma . \end{array} \ \ \ \ \ (3)

 

Here {k = 0 } or {\pm \kappa} is the sectional curvature of {(M,g)}. Moreover, equality in (3) holds if and only if {\eta} is the restriction of a function

\displaystyle u = \begin{cases} a_0 + \sum_{i=1}^n a_i x_i, & \displaystyle \mathrm{if} \ (M, g) =\mathbb{R}^n,\\ a_0 t + \sum_{i=1}^n a_i x_i, & \displaystyle \mathrm{if} \ (M, g) =\mathbb{H}^n(\kappa), \\ a_0 x_0 + \sum_{i=1}^n a_i x_i, & \displaystyle \mathrm{if} \ (M, g) =\mathbb{S}^n_+(\kappa) . \end{cases} \ \ \ \ \ (4)

 

Here {a_0, \ldots, a_n } are arbitrary constants, {\mathbb H^n (\kappa)} is identified with

\displaystyle \left\{ (t, x_1, \ldots, x_n) \in \mathbb R^{n,1} \ | \ - t^2 + \sum_{i=1}^n x_i^2 = - \frac{1}{\kappa}, \ t > 0 \right\}

in the {(n+1)}-dimensional Minkowski space {\mathbb R^{n,1}} and {\mathbb{S}^n_+ ( \kappa )} is identified with

\displaystyle \left\{ (x_0, x_1, \ldots, x_n) \in \mathbb R^{n+1} \ | \sum_{i=0}^n x_i^2 = \frac{1}{\kappa} , \ x_0 > 0 \right\}

in the {(n+1)}-dimensional Euclidean space {\mathbb R^{n+1}}.

Continue reading

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An extension of the First Fundamental Theorem of Calculus

In this note, we record a simple extension of the first fundamental theorem of calculus. Let us recall the first and second fundamental theorems of calculus:

Theorem 1 (First fundamental theorem of calculus) If {f:[a, b]\rightarrow \mathbb R} is continuous, then for {x\in (a, b),}

\displaystyle \frac{d}{dx}\int_a^x f(t)dt= f(x).

 

Theorem 2 (Second fundamental theorem of calculus) If {F:[a, b]\rightarrow \mathbb R} is {C^1}, then

\displaystyle \int_a^b F'(t)dt=F(b)-F(a).

 

It is well-known that Theorem 1 implies Theorem 2. Indeed, assuming Theorem 1, and for {F} in the assumption of Theorem 2, by letting {G(x)= \int_a^x F'(t)dt} and using Theorem 1 to differentiate the difference {G(x)-F(x)}, we can easily get Theorem 2.

It is also well-known that Theorem 2 can be generalized to many situations (usually called Stokes theorem or divergence theorem). E.g. for a smooth {(n-1)} form {\omega} on a compact manifold {M} with boundary {\partial M}, we have

\displaystyle \int_M d\omega = \int_{\partial M}\iota^*\omega.

Here {\iota} is the inclusion of {\partial M} in {M}.

It is therefore natural to ask if an extension of Theorem 1 exists in higher dimension. I will illustrate one such extension.

For simplicity let’s assume {f:\overline \Omega\rightarrow \mathbb R} is continuous and {h: \overline \Omega\rightarrow \mathbb R} is {C^1}, where {\Omega} is a bounded open set in {\mathbb R^n} and {\overline \Omega} is its closure. Then we have

Theorem 3 If {H_{n-1}(h^{-1}(t))<\infty}, then

\displaystyle \frac{d}{dt}\int_{\{x\in \Omega: h(x)\le t\}} f(x)|\nabla h(x)|dx= \int_{h^{-1}(t)}f(x)dH_{n-1}(x),

where {H_{n-1}} is the {(n-1)}-dimensional Hausdorff measure.  

When {n=1}, {\Omega=(a, b)} and {h(t)=t}, we can recover Theorem 1.
Proof: We have the coarea formula (which generalizes the Fubini’s theorem)

\displaystyle \int_\Omega f(x)|\nabla h(x)|dx= \int_{-\infty}^\infty \int_{h^{-1}(t)} f(x)dH_{n-1}(x)dt.

Apply this formula to {\Omega_s =\{x\in \Omega: h(x)\le s\}}, then we have

\displaystyle \int_{\{x\in \Omega: h(x)\le s\}}f(x)|\nabla h(x)|dx= \int_{-\infty}^s \int_{h^{-1}(t)}f(x)dH_{n-1}(x)dt.

Differentiating both sides with respect to {s}, and applying the first fundamental theorem of calculus (Theorem 1), we can get the result. \Box

Problem: Find the most general settings for this result to hold. Do we have other non-trivial generalization?

Posted in Analysis, Calculus | Leave a comment