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Posted in Miscellaneous | 2 Comments

Why is a² + b² ≥ 2ab ?

This post can be regarded as a sequel to my previous (and very ancient) post on 1+2+3+…. Though these two posts are not quite logically related, they share the same spirit (I’m asking a dumb question again).

How can one prove the following?

Theorem 1

\displaystyle \begin{array}{rl} \displaystyle a^2+b^2\ge 2ab.\end{array}

This extremely simple inequality (just next to the simplest but arguably the most important inequality {a^2\ge 0}) turns out to be the testing ground for a lot of more advanced inequalities (e.g. Cauchy-Schwarz, or Newton’s inequality).

In this post, I record some proofs of this inequality that I can think of. Some of them are quite similar, or may be considered almost the same depending how you measure the level of similarity. I try to group proofs of roughly the same idea together. In some of these proofs, we have actually used more advanced inequalities (which perhaps are even proved from this simple inequality). For such a simple result, I guess there may be over a hundred proofs.

So what’s the point of doing all these, since only one proof is all that suffices? Because not many people would have the patience to read till the end (or even this paragraph), let me put the summary here:

Don’t just read it; fight it! Ask your own questions, look for your own examples, discover your own proofs. Is the hypothesis necessary? Is the converse true? What happens in the classical special case? What about the degenerate cases? Where does the proof use the hypothesis? (Paul Halmos, “I want to be a mathematician: an automathography”)

So, let’s begin.

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Posted in Geometry, Probability, Linear Algebra, Discrete Mathematics, Calculus, Inequalities | 1 Comment

A remark on the divergence theorem

The divergence theorem states that for a compact domain {D} in {\mathbb R^3} with piecewise smooth boundary {\partial D}, then for a smooth vector field {\textbf{F}} on {D}, we have

\displaystyle \begin{array}{rl} \displaystyle \iint_{\partial D}\textbf{F}\cdot \textbf{n}\,dS=\iiint_D \nabla \cdot\textbf{F}\,dV,\end{array}

where {\textbf{n}} is the unit outward normal and {\nabla \cdot\textbf{F}} is the divergence of {\textbf{F}}.

In most textbooks, the divergence theorem is proved by the following strategy:

  1. Prove that

    \displaystyle \begin{array}{rl} \displaystyle  \iint _{\partial D}(0, 0, R)\cdot \textbf{n} \;dS= \iiint _D\frac{\partial R}{\partial z}\;dV \ \ \ \ \ (1)\end{array}

    where {D} is a so called type {1} domain, i.e. {D=\{(x, y, z): (x, y)\in \Omega, \underline f(x, y)\le z\le \bar f(x, y)\}} for some region {\Omega\subset \mathbb R^2} and {\bar f, \underline f\in C^1(R)}. i.e. {D} is bounded by the graphs of two functions with variables {x, y}. This can be quite easily done using the fundamental theorem of calculus.

  2. Similarly prove

    \displaystyle \begin{array}{rl} \displaystyle  \iint _{\partial D}(P, 0, 0)\cdot \textbf{n} \;dS= \iiint _D\frac{\partial P}{\partial x}\;dV \ \ \ \ \ (2)\end{array}

    and

    \displaystyle \begin{array}{rl} \displaystyle  \iint _{\partial D}(0, Q, 0)\cdot \textbf{n} \;dS= \iiint _D\frac{\partial Q}{\partial y}\;dV \ \ \ \ \ (3)\end{array}

    for the so called type {2} domain and type {3} domain respectively.

  3. Argue that for sufficiently nice {D}, it can be cut into finitely many pieces {D_i}, each of which are of both type {1}, {2}, and {3} (the so called type {4} domain). Thus we can compute the divergence integral {\displaystyle \iiint_D \nabla \cdot \textbf{F}\,dV} by adding the integrals of the pieces and apply the previous result (using linearity). Argue that the orientation of the common face of these {D_i} (if any) is opposite to each other and thus a cancellation argument will give the surface integral {\displaystyle \iint _{\partial D}\textbf{F}\cdot \textbf{n}\;dS}.

I am slightly disappointed by this approach because even for the so called type {1} domain, we still cannot prove the divergence theorem very directly (say, by reducing to a double integral on some planar domain {R} and apply Green’s theorem), but have to further cut it into even smaller type {4} domains.

In this note we modify the strategy above by proving (1), (2), (3) all hold on a type {1} domain. In particular, I will show that (2) and (3) can be proved for a type {1} domain by using Green’s theorem.
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Posted in Calculus | Leave a comment

The Cauchy-Schwarz inequality and the Lagrange identity

The classical Lagrange identity is the following:

\displaystyle \begin{array}{rl} \displaystyle \left(\sum_{i=1}^n a_ib_i\right)^2+\sum _{1\le i<j\le n} \left(a_ib_j-a_jb_i\right)^2 =\left(\sum_{i=1}^{n}a_i^2\right)\left(\sum_{j=1}^{n}b_j^2\right).\end{array}

This can be proven by expanding {\displaystyle \sum _{1\le i<j\le n}\left(a_ib_j-a_jb_i\right)^2 } and separating the terms into the cross-terms part and the non cross-terms part.

The Lagrange identity implies the Cauchy-Schwarz inequality in {\mathbb R^n}. And when {n=3}, this can be rephrased as

\displaystyle \begin{array}{rl} \displaystyle |{a}|^2|{b}|^2= \left({a}\cdot {b}\right) ^2+|{a}\times {b}|^2\end{array}

for {{a}, {b}\in \mathbb R^3}. In general, the term {\displaystyle \sum _{1\le i<j\le n} \left(a_ib_j-a_jb_i\right)^2} can be identified as the norm squared of the wedge product {{a}\wedge {b}}.

In this note, we give the less well-known extension of this identity and the corresponding Cauchy-Schwarz type inequality.
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Posted in Algebra, Group theory, Inequalities, Linear Algebra | Leave a comment

On the existence of a metric compatible with a given connection

Question: Suppose we are given a torsion-free (i.e. the torsion tensor vanishes) affine connection {\nabla } on a smooth connected manifold {M}. Does there exist a Riemannian metric {g} such that its Levi-Civita connection is {\nabla }? If so, is it unique if we prescribe its value at a point?

Let {(x^i)} be a local coordinates on {M} and let {\nabla _i\partial _j=\Gamma_{ij}^k\partial _k} (using Einstein’s notation). The torsion-free condition is then equivalent to {\Gamma_{ij}^k=\Gamma_{ji}^k}. We want to solve the linear system of partial differential equations (Note that {\Gamma_{ij}^k} are {C^1} as {\nabla } is a connection.)

\displaystyle \begin{array}{rl} \displaystyle  \partial _k g_{ij}=\Gamma_{ki}^pg_{pj}+\Gamma_{kj}^pg_{pi}. \ \ \ \ \ (1)\end{array}

We impose an initial condition at {x_0} by

\displaystyle \begin{array}{rl} \displaystyle g_{ij}(x_0)= h_{ij}\end{array}

for some fixed symmetric {h_{ij}}.

It is clear from the initial condition and from (1) that {g_{ij}} is symmetric if the solution exists. According to the theory of first order partial differential equations system (cf. e.g. Stoker’s Differential Geometry Appendix B), the system (1) is uniquely solvable if and only if the compatibility conditions hold:

\displaystyle \begin{array}{rl} \displaystyle   & \displaystyle \partial _l\Gamma_{ki}^pg_{pj} +\Gamma_{ki}^p \partial _lg_{pj} +\partial _l\Gamma_{kj}^pg_{ip} +\Gamma_{kj}^p \partial _lg_{ip}\\ =& \displaystyle \partial _k\Gamma_{li}^pg_{pj} +\Gamma_{li}^p \partial _kg_{pj} +\partial _k\Gamma_{lj}^pg_{ip} +\Gamma_{lj}^p \partial _kg_{ip}.  \ \ \ \ \ (2)\end{array}

In view of (1), this is equivalent to

\displaystyle \begin{array}{rl} \displaystyle   & \displaystyle \partial _l\Gamma_{ki}^pg_{pj} +\Gamma_{ki}^p (\Gamma_{lp}^m g_{mj}+\Gamma_{lj}^m g_{pm}) +\partial _l\Gamma_{kj}^pg_{ip} +\Gamma_{kj}^p (\Gamma_{li}^m g_{mp}+\Gamma_{lp}^m g_{im})\\ =& \displaystyle \partial _k\Gamma_{li}^pg_{pj} +\Gamma_{li}^p (\Gamma_{kp}^m g_{mj}+\Gamma_{kj}^m g_{pm}) +\partial _k\Gamma_{lj}^pg_{ip} +\Gamma_{lj}^p (\Gamma_{ki}^m g_{mp}+\Gamma_{kp}^m g_{im}).  \ \ \ \ \ (3)\end{array}

To shed some light on the following computation, let us introduce the curvature tensor associated with {\nabla }.

Definition 1 Given a connection {\nabla } and vector fields {X}, {Y}, {Z}, we define the curvature tensor {R(X, Y)Z} by

\displaystyle \begin{array}{rl} \displaystyle  R (X, Y)Z=\nabla _X\nabla _YZ-\nabla _Y\nabla _XZ-\nabla _{[X,Y]}Z. \ \ \ \ \ (4)\end{array}

Here {[\cdot,\cdot]} is the Lie bracket of vector fields. It is readily checked that {R(\cdot, \cdot)\cdot} is a tensor field regardless of whether {\nabla } is torsion-free or not. In local coordinates, we define {R_{ijk}^l} by

\displaystyle \begin{array}{rl} \displaystyle R(\partial _i, \partial _j)\partial _k=R_{ijk}^l\partial _l.\end{array}

It is not hard to see that if {\nabla } is torsion-free and the {\nabla } is compatible with {g} in the sense of (1), then we have the local formula

\displaystyle \begin{array}{rl} \displaystyle R_{ijk}^l = \partial _i \Gamma_{jk}^l - \partial _j \Gamma_{ik }^l + \Gamma _{jk}^m \Gamma_{im}^l- \Gamma _{ik}^m \Gamma _{jm}^l.\end{array}

We subtract RHS from LHS of (3) and get

\displaystyle \begin{array}{rl} \displaystyle   & \displaystyle \partial _l\Gamma_{ki}^pg_{pj} +\Gamma_{ki}^p (\Gamma_{lp}^m g_{mj}+\Gamma_{lj}^m g_{pm}) +\partial _l\Gamma_{kj}^pg_{ip} +\Gamma_{kj}^p (\Gamma_{li}^m g_{mp}+\Gamma_{lp}^m g_{im})\\ & \displaystyle -\left(\partial _k\Gamma_{li}^pg_{pj} +\Gamma_{li}^p (\Gamma_{kp}^m g_{mj}+\Gamma_{kj}^m g_{pm}) +\partial _k\Gamma_{lj}^pg_{ip} +\Gamma_{lj}^p (\Gamma_{ki}^m g_{mp}+\Gamma_{kp}^m g_{im})\right)\\ =& \displaystyle  \left(\partial _l \Gamma_{ki}^m-\partial _k \Gamma_{li}^m+\Gamma_{ki}^p\Gamma_{lp}^m-\Gamma_{li}^p\Gamma_{kp}^m\right)g_{mj} +\left(\partial _l \Gamma_{kj}^m-\partial _k \Gamma_{lj}^m+\Gamma_{kj}^p\Gamma_{lp}^m-\Gamma_{lj}^p\Gamma_{kp}^m\right)g_{mi}\\ & \displaystyle +\Gamma_{ki}^p\Gamma_{lj}^m g_{pm}+\Gamma_{kj}^p\Gamma_{li}^m g_{mp}-\Gamma_{li}^p \Gamma_{kj}^m g_{pm}-\Gamma_{lj}^p\Gamma_{ki}^m g_{mp}\\ =& \displaystyle  \left(\partial _l \Gamma_{ki}^m-\partial _k \Gamma_{li}^m+\Gamma_{ki}^p\Gamma_{lp}^m-\Gamma_{li}^p\Gamma_{kp}^m\right)g_{mj} +\left(\partial _l \Gamma_{kj}^m-\partial _k \Gamma_{lj}^m+\Gamma_{kj}^p\Gamma_{lp}^m-\Gamma_{lj}^p\Gamma_{kp}^m\right)g_{mi}\\ =& \displaystyle R_{lki}^m g_{mj}+R_{lkj}^m g_{mi}.  \ \ \ \ \ (5)\end{array}

Let us show that (1) implies {R_{lki}^m g_{mj}+R_{lkj}^m g_{mi}=0}, so by (5), we conclude that the compatibility condition (2) holds. To see this, using the compatibility of metric (1) (We do not sum over {k} in the following):

\displaystyle \begin{array}{rl} \displaystyle   \frac{1}{2}\partial _i\partial _j g_{kk} =\partial _i\left(\Gamma_{jk}^l g_{lk}\right) =\partial _i\Gamma_{jk}^l g_{lk}+\Gamma_{jk}^l\Gamma_{il}^m g_{mk} +\Gamma_{jk}^l\Gamma_{ik}^m g_{ml} \end{array}

and similarly

\displaystyle \begin{array}{rl} \displaystyle \frac{1}{2}\partial _j \partial _i g_{kk} =\partial _j\Gamma_{ik}^l g_{lk}+\Gamma_{ik}^l\Gamma_{jl}^m g_{mk} +\Gamma_{ik}^l\Gamma_{jk}^m g_{ml}. \end{array}

Subtracting the two equations,

\displaystyle \begin{array}{rl} \displaystyle   0 =\partial _i\Gamma_{jk}^l g_{lk}-\partial _j\Gamma_{ik}^l g_{lk}+\Gamma_{jk}^l\Gamma_{il}^m g_{mk} -\Gamma_{ik}^l\Gamma_{jl}^m g_{mk} = R_{ijk}^lg_{lk}. \end{array}

This implies that {g(R(X, Y)Z, Z)=0} if {g} is defined by {g:=g_{ij}dx^idx^j}. This is easily seen to be equivalent to {R_{lki}^m g_{mj}+R_{lkj}^m g_{mi}=0}. We conclude that (2) holds, and therefore we have proved the existence and uniqueness of the solution to the system

\displaystyle \begin{array}{rl} \displaystyle   \begin{cases} g_{ij,k}=\Gamma_{ki}^pg_{pj}+\Gamma_{kj}^pg_{ip}\\ g_{ij}(x_0)=h_{ij}. \end{cases} \end{array}

Gometrically, if we prescribe an inner product on {T_{p_0}M} for some {p_0\in M}, then there is a unique Riemannian metric which is compatible with the torsion-free connection {\nabla }. So we have proved

Theorem 2 Suppose {\nabla } is a torsion-free affine connection on a smooth connected manifold {M}. Let {g_0} be an inner product on {T_{p_0}M}, where {p_0\in M}. Then there exists a unique Riemannian metric {g} such that {\nabla } is the Levi-Civita connection of {g} and {g(p_0)=g_0}.

Posted in Differential equations, Geometry | 1 Comment

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

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.

 

Posted in Algebraic geometry | Leave a comment

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

Posted in Calculus, General Relativity, Geometry, Inequalities | Leave a comment