The Legend about the importance of Amarnath Cave is as follows : -
This is The Cave which was chosen by "Bhole Shankar" for narrating the secrets of immortality and creation of Universe to Maa Parvati ji.

The Amar Kathaa :
Centuries ago Maa Parvati asked Shiv ji to let her know why and when He started wearing the beads of heads ( Mund Mala) . Bhole Shankar replied when ever you are born I add one more head in my beads . Maa Parvati said ," My Lord, my body is destroyed every time and I die again and again, but you are Immortal. Please let me know the Secret of this". Bhole Shankar replied that it is due to Amar Katha .
Maa Parvati insisted that she may be told that secret. For long Shiva ji continued postponing . Finally on consistent demand from Maa Parvati He made up his mind to tell the immortal secret . He started for lonely place where no living being could listen it . He choose Amarnath Cave . In preparation to that He left His Nandi (The Bull which He used to ride) at Pahalgam (formar name Bail Gaon) . At Chandanwari He released Moon from his hairs (Jataon). At the banks of Lake Sheshnag He released the Sheshnag snakes. He decided to leave his Son Ganesha at Mahagunas Parvat (Mahaganesh Hill) . At Panjtarni, Shivji left the Five Elements behind (Earth , Water, Air , Fire and Sky) which make living being . He is the Lord of these elements. It is believed that as a symbol of sacrificing the earthly world , Shivaji and Maa Parvati had Tandav Dance . After leaving behind all these, Bhole Shankar enters the Holy Amarnath Cave along with Parvati Maa.
Lord Shiva takes his Samadhi on the Deer Skin and concentrate . To ensure that no living being is able to hear the Immortal Tale , He created Rudra named Kalagni and ordered him to spread fire to eliminate every living thing in and around the Holy Cave . After this He started narrating the secret of Amarnath immortality to Maa Parvati . But as a matter of chance one egg which was lying beneath the Deer skin remained protected . It is believed to be non living and more over it was protected by Shiva -Parvati Asan (Bed) . The pair of pigeons which were born out of this egg became immortal having listened the secret of immortality (Amar Katha). Many pilgrims report seeing the pair of pigeons when they trek the arduous route to pay their obeisance before the Ice-Lingam.